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Friable values of polynomials Greg Martin Introduction Friable - PowerPoint PPT Presentation

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Friable values of polynomials Friable values of polynomials Greg Martin Introduction Friable integers How often do the values of a polynomial Friable values of polynomials have only small prime factors? Bounds for friable values of

  1. Friable values of The Dickman–de Bruijn ρ -function polynomials Greg Martin Definition Introduction Friable integers ρ ( u ) is the unique continuous solution of the Friable values of polynomials differential-difference equation u ρ ′ ( u ) = − ρ ( u − 1 ) for u ≥ 1 Bounds for friable values of polynomials that satisfies the initial condition ρ ( u ) = 1 for 0 ≤ u ≤ 1. How friable can values of special polynomials be? How friable can values of general polynomials be? Example Can we have lots of friable values? Conjecture for prime For 1 ≤ u ≤ 2, values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) ρ ′ ( u ) = − ρ ( u − 1 ) = − 1 A uniform version of Hypothesis H u = ⇒ ρ ( u ) = C − log u . Conjecture for friable u values of polynomials Statement of the conjecture Since ρ ( u ) = 1, we have ρ ( u ) = 1 − log u for 1 ≤ u ≤ 2. Reduction to convenient polynomials Translation into prime values of 2 when u = √ e . Therefore polynomials Consequence: Note that ρ ( u ) = 1 Shepherding the local factors Sums of multiplicative functions the “median size” of the largest prime factor Summary of n is n 1 / √ e .

  2. Friable values of Friable numbers among values of polynomials polynomials Greg Martin Introduction Definition Friable integers Friable values of polynomials Ψ( F ; x , y ) is the number of integers n up to x such that all Bounds for friable values of polynomials the prime factors of F ( n ) are all at most y : How friable can values of special polynomials be? How friable can values of general Ψ( F ; x , y ) = # { 1 ≤ n ≤ x : p | F ( n ) = ⇒ p ≤ y } polynomials be? Can we have lots of friable values? Conjecture for prime values of polynomials When F ( x ) is a linear polynomial (friable numbers in Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) arithmetic progressions), we have the same A uniform version of Hypothesis H � log x � asymptotic Ψ( F ; x , y ) ∼ ρ . Conjecture for friable log y values of polynomials Knowing the size of Ψ( F ; x , y ) has applications to Statement of the conjecture Reduction to convenient polynomials analyzing the running time of modern factoring Translation into prime values of algorithms (quadratic sieve, number field sieve). polynomials Shepherding the local factors Sums of multiplicative functions A basic sort of question in number theory: are two Summary arithmetic properties (in this case, friability and being the value of a polynomial) independent?

  3. Friable values of Friable numbers among values of polynomials polynomials Greg Martin Introduction Definition Friable integers Friable values of polynomials Ψ( F ; x , y ) is the number of integers n up to x such that all Bounds for friable values of polynomials the prime factors of F ( n ) are all at most y : How friable can values of special polynomials be? How friable can values of general Ψ( F ; x , y ) = # { 1 ≤ n ≤ x : p | F ( n ) = ⇒ p ≤ y } polynomials be? Can we have lots of friable values? Conjecture for prime values of polynomials When F ( x ) is a linear polynomial (friable numbers in Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) arithmetic progressions), we have the same A uniform version of Hypothesis H � log x � asymptotic Ψ( F ; x , y ) ∼ ρ . Conjecture for friable log y values of polynomials Knowing the size of Ψ( F ; x , y ) has applications to Statement of the conjecture Reduction to convenient polynomials analyzing the running time of modern factoring Translation into prime values of algorithms (quadratic sieve, number field sieve). polynomials Shepherding the local factors Sums of multiplicative functions A basic sort of question in number theory: are two Summary arithmetic properties (in this case, friability and being the value of a polynomial) independent?

  4. Friable values of Friable numbers among values of polynomials polynomials Greg Martin Introduction Definition Friable integers Friable values of polynomials Ψ( F ; x , y ) is the number of integers n up to x such that all Bounds for friable values of polynomials the prime factors of F ( n ) are all at most y : How friable can values of special polynomials be? How friable can values of general Ψ( F ; x , y ) = # { 1 ≤ n ≤ x : p | F ( n ) = ⇒ p ≤ y } polynomials be? Can we have lots of friable values? Conjecture for prime values of polynomials When F ( x ) is a linear polynomial (friable numbers in Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) arithmetic progressions), we have the same A uniform version of Hypothesis H � log x � asymptotic Ψ( F ; x , y ) ∼ ρ . Conjecture for friable log y values of polynomials Knowing the size of Ψ( F ; x , y ) has applications to Statement of the conjecture Reduction to convenient polynomials analyzing the running time of modern factoring Translation into prime values of algorithms (quadratic sieve, number field sieve). polynomials Shepherding the local factors Sums of multiplicative functions A basic sort of question in number theory: are two Summary arithmetic properties (in this case, friability and being the value of a polynomial) independent?

  5. Friable values of polynomials Greg Martin Introduction Introduction 1 Friable integers Friable values of polynomials Bounds for friable Bounds for friable values of polynomials 2 values of polynomials How friable can values of special polynomials be? How friable can values of special polynomials be? How friable can values of general polynomials be? How friable can values of general polynomials be? Can we have lots of friable values? Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” 3 Conjecture for prime values of polynomials (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable 4 Conjecture for friable values of polynomials values of polynomials Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  6. Friable values of How friable can values of special polynomials polynomials be? Greg Martin Introduction Friable integers Friable values of polynomials For binomials, there’s a nice trick which yields: Bounds for friable values of polynomials Theorem (Schinzel, 1967) How friable can values of special polynomials be? How friable can values of general For any nonzero integers A and B, any positive integer d, polynomials be? Can we have lots of friable values? and any ε > 0 , there are infinitely many numbers n for which An d + B is n ε -friable. Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Balog and Wooley (1998), building on an idea of A uniform version of Hypothesis H Conjecture for friable Eggleton and Selfridge, extended this result to values of polynomials products of binomials Statement of the conjecture Reduction to convenient polynomials L Translation into prime values of polynomials ( A j n d j + B j ) . � Shepherding the local factors Sums of multiplicative functions j = 1 Summary

  7. Friable values of How friable can values of special polynomials polynomials be? Greg Martin Introduction Friable integers Friable values of polynomials For binomials, there’s a nice trick which yields: Bounds for friable values of polynomials Theorem (Schinzel, 1967) How friable can values of special polynomials be? How friable can values of general For any nonzero integers A and B, any positive integer d, polynomials be? Can we have lots of friable values? and any ε > 0 , there are infinitely many numbers n for which An d + B is n ε -friable. Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Balog and Wooley (1998), building on an idea of A uniform version of Hypothesis H Conjecture for friable Eggleton and Selfridge, extended this result to values of polynomials products of binomials Statement of the conjecture Reduction to convenient polynomials L Translation into prime values of polynomials ( A j n d j + B j ) . � Shepherding the local factors Sums of multiplicative functions j = 1 Summary

  8. Friable values of Proof for an explicit binomial polynomials Greg Martin Example Introduction For any ε > 0, there are infinitely many numbers n for which Friable integers F ( n ) = 3 n 5 + 7 is n ε -friable. Friable values of polynomials Bounds for friable values of polynomials Define n k = 3 8 k − 1 7 2 k . Then How friable can values of special polynomials be? How friable can values of general F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 polynomials be? � � Can we have lots of friable values? Conjecture for prime values of polynomials factors into values of cyclotomic polynomials: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 Conjecture for friable values of polynomials m | ( 10 k − 1 ) Statement of the conjecture Reduction to convenient polynomials Translation into prime values of � x − e 2 π ir / m � polynomials � Φ m ( x ) = Shepherding the local factors Sums of multiplicative functions 1 ≤ r ≤ m ( r , m )= 1 Summary Φ m has integer coefficients and degree φ ( m )

  9. Friable values of Proof for an explicit binomial polynomials Greg Martin Example Introduction For any ε > 0, there are infinitely many numbers n for which Friable integers F ( n ) = 3 n 5 + 7 is n ε -friable. Friable values of polynomials Bounds for friable values of polynomials Define n k = 3 8 k − 1 7 2 k . Then How friable can values of special polynomials be? How friable can values of general F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 polynomials be? � � Can we have lots of friable values? Conjecture for prime values of polynomials factors into values of cyclotomic polynomials: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 Conjecture for friable values of polynomials m | ( 10 k − 1 ) Statement of the conjecture Reduction to convenient polynomials Translation into prime values of � x − e 2 π ir / m � polynomials � Φ m ( x ) = Shepherding the local factors Sums of multiplicative functions 1 ≤ r ≤ m ( r , m )= 1 Summary Φ m has integer coefficients and degree φ ( m )

  10. Friable values of Proof for an explicit binomial polynomials Greg Martin Example Introduction For any ε > 0, there are infinitely many numbers n for which Friable integers F ( n ) = 3 n 5 + 7 is n ε -friable. Friable values of polynomials Bounds for friable values of polynomials Define n k = 3 8 k − 1 7 2 k . Then How friable can values of special polynomials be? How friable can values of general F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 polynomials be? � � Can we have lots of friable values? Conjecture for prime values of polynomials factors into values of cyclotomic polynomials: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 Conjecture for friable values of polynomials m | ( 10 k − 1 ) Statement of the conjecture Reduction to convenient polynomials Translation into prime values of � x − e 2 π ir / m � polynomials � Φ m ( x ) = Shepherding the local factors Sums of multiplicative functions 1 ≤ r ≤ m ( r , m )= 1 Summary Φ m has integer coefficients and degree φ ( m )

  11. Friable values of Proof for an explicit binomial polynomials Greg Martin Example Introduction For any ε > 0, there are infinitely many numbers n for which Friable integers F ( n ) = 3 n 5 + 7 is n ε -friable. Friable values of polynomials Bounds for friable values of polynomials Define n k = 3 8 k − 1 7 2 k . Then How friable can values of special polynomials be? How friable can values of general F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 polynomials be? � � Can we have lots of friable values? Conjecture for prime values of polynomials factors into values of cyclotomic polynomials: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 Conjecture for friable values of polynomials m | ( 10 k − 1 ) Statement of the conjecture Reduction to convenient polynomials Translation into prime values of � x − e 2 π ir / m � polynomials � Φ m ( x ) = Shepherding the local factors Sums of multiplicative functions 1 ≤ r ≤ m ( r , m )= 1 Summary Φ m has integer coefficients and degree φ ( m )

  12. Friable values of Proof for an explicit binomial polynomials Greg Martin Example Introduction For any ε > 0, there are infinitely many numbers n for which Friable integers F ( n ) = 3 n 5 + 7 is n ε -friable. Friable values of polynomials Bounds for friable values of polynomials Define n k = 3 8 k − 1 7 2 k . Then How friable can values of special polynomials be? How friable can values of general F ( n k ) = 3 5 ( 8 k − 1 )+ 1 7 5 ( 2 k ) + 7 = − 7 ( − 3 4 7 ) 10 k − 1 − 1 polynomials be? � � Can we have lots of friable values? Conjecture for prime values of polynomials factors into values of cyclotomic polynomials: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H � Φ m ( − 3 4 7 ) . F ( n k ) = − 7 Conjecture for friable values of polynomials m | ( 10 k − 1 ) Statement of the conjecture Reduction to convenient polynomials Translation into prime values of � x − e 2 π ir / m � polynomials � Φ m ( x ) = Shepherding the local factors Sums of multiplicative functions 1 ≤ r ≤ m ( r , m )= 1 Summary Φ m has integer coefficients and degree φ ( m )

  13. Friable values of polynomials Greg Martin From the last slide Introduction F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) Friable integers F ( n k ) = − 7 Friable values of polynomials m | ( 10 k − 1 ) Bounds for friable n k = 3 8 k − 1 7 2 k values of polynomials How friable can values of special polynomials be? How friable can values of general � � Φ m ( − 3 4 7 ) � primes dividing F ( n k ) are ≤ max polynomials be? � Can we have lots of friable values? m | ( 10 k − 1 ) Conjecture for prime Φ m ( x ) is roughly x φ ( m ) ≤ x φ ( 10 k − 1 ) values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of A uniform version of Hypothesis H F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) Conjecture for friable values of polynomials Statement of the conjecture infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Reduction to convenient polynomials Translation into prime values of How many such friable values? ≫ F ,ε log x , for n ≤ x polynomials Shepherding the local factors Sums of multiplicative functions ε can be made quantitative n c F / log log log n -friable values Summary

  14. Friable values of polynomials Greg Martin From the last slide Introduction F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) Friable integers F ( n k ) = − 7 Friable values of polynomials m | ( 10 k − 1 ) Bounds for friable n k = 3 8 k − 1 7 2 k values of polynomials How friable can values of special polynomials be? How friable can values of general � � Φ m ( − 3 4 7 ) � primes dividing F ( n k ) are ≤ max polynomials be? � Can we have lots of friable values? m | ( 10 k − 1 ) Conjecture for prime Φ m ( x ) is roughly x φ ( m ) ≤ x φ ( 10 k − 1 ) values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of A uniform version of Hypothesis H F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) Conjecture for friable values of polynomials Statement of the conjecture infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Reduction to convenient polynomials Translation into prime values of How many such friable values? ≫ F ,ε log x , for n ≤ x polynomials Shepherding the local factors Sums of multiplicative functions ε can be made quantitative n c F / log log log n -friable values Summary

  15. Friable values of polynomials Greg Martin From the last slide Introduction F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) Friable integers F ( n k ) = − 7 Friable values of polynomials m | ( 10 k − 1 ) Bounds for friable n k = 3 8 k − 1 7 2 k values of polynomials How friable can values of special polynomials be? How friable can values of general � � Φ m ( − 3 4 7 ) � primes dividing F ( n k ) are ≤ max polynomials be? � Can we have lots of friable values? m | ( 10 k − 1 ) Conjecture for prime Φ m ( x ) is roughly x φ ( m ) ≤ x φ ( 10 k − 1 ) values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of A uniform version of Hypothesis H F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) Conjecture for friable values of polynomials Statement of the conjecture infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Reduction to convenient polynomials Translation into prime values of How many such friable values? ≫ F ,ε log x , for n ≤ x polynomials Shepherding the local factors Sums of multiplicative functions ε can be made quantitative n c F / log log log n -friable values Summary

  16. Friable values of polynomials Greg Martin From the last slide Introduction F ( n ) = 3 n 5 + 7 � Φ m ( − 3 4 7 ) Friable integers F ( n k ) = − 7 Friable values of polynomials m | ( 10 k − 1 ) Bounds for friable n k = 3 8 k − 1 7 2 k values of polynomials How friable can values of special polynomials be? How friable can values of general � � Φ m ( − 3 4 7 ) � primes dividing F ( n k ) are ≤ max polynomials be? � Can we have lots of friable values? m | ( 10 k − 1 ) Conjecture for prime Φ m ( x ) is roughly x φ ( m ) ≤ x φ ( 10 k − 1 ) values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) n k is roughly ( 3 4 7 ) 4 k , but the largest prime factor of A uniform version of Hypothesis H F ( n k ) is bounded by roughly ( 3 4 7 ) φ ( 10 k − 1 ) Conjecture for friable values of polynomials Statement of the conjecture infinitely many k with φ ( 10 k − 1 ) / 4 k < ε Reduction to convenient polynomials Translation into prime values of How many such friable values? ≫ F ,ε log x , for n ≤ x polynomials Shepherding the local factors Sums of multiplicative functions ε can be made quantitative n c F / log log log n -friable values Summary

  17. Friable values of Polynomial factorizations polynomials Greg Martin Example Introduction The polynomial F ( x + F ( x )) is always divisible by F ( x ) . In Friable integers particular, if deg F = d , then F ( x + F ( x )) is roughly x d 2 yet Friable values of polynomials Bounds for friable is automatically roughly x d 2 − d -friable. values of polynomials How friable can values of special polynomials be? How friable can values of general polynomials be? Mnemonic Can we have lots of friable values? Conjecture for prime x + F ( x ) ≡ x (mod F ( x ) ) values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Special case: A uniform version of Hypothesis H Conjecture for friable If F ( x ) is quadratic with lead coefficient a , then values of polynomials Statement of the conjecture Reduction to convenient x + 1 � � F ( x + F ( x )) = F ( x ) · aF . polynomials a Translation into prime values of polynomials Shepherding the local factors In particular, if F ( x ) = x 2 + bx + c , then Sums of multiplicative functions Summary F ( x + F ( x )) = F ( x ) F ( x + 1 ) .

  18. Friable values of Polynomial factorizations polynomials Greg Martin Example Introduction The polynomial F ( x + F ( x )) is always divisible by F ( x ) . In Friable integers particular, if deg F = d , then F ( x + F ( x )) is roughly x d 2 yet Friable values of polynomials Bounds for friable is automatically roughly x d 2 − d -friable. values of polynomials How friable can values of special polynomials be? How friable can values of general polynomials be? Mnemonic Can we have lots of friable values? Conjecture for prime x + F ( x ) ≡ x (mod F ( x ) ) values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Special case: A uniform version of Hypothesis H Conjecture for friable If F ( x ) is quadratic with lead coefficient a , then values of polynomials Statement of the conjecture Reduction to convenient x + 1 � � F ( x + F ( x )) = F ( x ) · aF . polynomials a Translation into prime values of polynomials Shepherding the local factors In particular, if F ( x ) = x 2 + bx + c , then Sums of multiplicative functions Summary F ( x + F ( x )) = F ( x ) F ( x + 1 ) .

  19. Friable values of A refinement of Schinzel polynomials Greg Martin Idea: use the reciprocal polynomial x d F ( 1 / x ) . Introduction Friable integers Restrict to F ( x ) = x d + a 2 x d − 2 + . . . for simplicity. Friable values of polynomials Bounds for friable values of polynomials How friable can values of special Proposition polynomials be? How friable can values of general polynomials be? Let h ( x ) be a polynomial such that xh ( x ) − 1 is divisible by Can we have lots of friable values? x d F ( 1 / x ) . Then F ( h ( x )) is divisible by x d F ( 1 / x ) . In Conjecture for prime values of polynomials particular, we can take deg h = d − 1 , in which case F ( h ( x )) Schinzel’s “Hypothesis H” is roughly x d 2 − d yet is automatically roughly x d 2 − 2 d -friable. (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials Mnemonic Statement of the conjecture Reduction to convenient polynomials h ( x ) ≡ 1 / x (mod F ( 1 / x ) ) Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Note: The proposition isn’t true for d = 2, since the leftover Summary “factor” of degree 2 2 − 2 · 2 = 0 is a constant.

  20. Friable values of A refinement of Schinzel polynomials Greg Martin Idea: use the reciprocal polynomial x d F ( 1 / x ) . Introduction Friable integers Restrict to F ( x ) = x d + a 2 x d − 2 + . . . for simplicity. Friable values of polynomials Bounds for friable values of polynomials How friable can values of special Proposition polynomials be? How friable can values of general polynomials be? Let h ( x ) be a polynomial such that xh ( x ) − 1 is divisible by Can we have lots of friable values? x d F ( 1 / x ) . Then F ( h ( x )) is divisible by x d F ( 1 / x ) . In Conjecture for prime values of polynomials particular, we can take deg h = d − 1 , in which case F ( h ( x )) Schinzel’s “Hypothesis H” is roughly x d 2 − d yet is automatically roughly x d 2 − 2 d -friable. (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials Mnemonic Statement of the conjecture Reduction to convenient polynomials h ( x ) ≡ 1 / x (mod F ( 1 / x ) ) Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Note: The proposition isn’t true for d = 2, since the leftover Summary “factor” of degree 2 2 − 2 · 2 = 0 is a constant.

  21. Friable values of Recursively use Schinzel’s construction polynomials Greg Martin D m : an unspecified polynomial of degree m Introduction Friable integers Friable values of polynomials Bounds for friable Example values of polynomials How friable can values of special polynomials be? deg F ( x ) = 4. Use Schinzel’s construction repeatedly: How friable can values of general polynomials be? Can we have lots of friable values? D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 Conjecture for prime D 84 = F ( D 21 ) = D 28 D 8 D 48 “score” = 16/7 values of polynomials Schinzel’s “Hypothesis H” D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 736/329 (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials For deg F = 2, begin with F ( D 4 ) = D 2 D 2 D 4 . Statement of the conjecture Specifically, Reduction to convenient polynomials Translation into prime values of polynomials � � �� � x + 1 � F x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . Shepherding the local factors a Sums of multiplicative functions Summary For deg F = 3, begin with F ( D 4 ) = D 3 D 3 D 6 .

  22. Friable values of Recursively use Schinzel’s construction polynomials Greg Martin D m : an unspecified polynomial of degree m Introduction Friable integers Friable values of polynomials Bounds for friable Example values of polynomials How friable can values of special polynomials be? deg F ( x ) = 4. Use Schinzel’s construction repeatedly: How friable can values of general polynomials be? Can we have lots of friable values? D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 Conjecture for prime D 84 = F ( D 21 ) = D 28 D 8 D 48 “score” = 16/7 values of polynomials Schinzel’s “Hypothesis H” D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 736/329 (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials For deg F = 2, begin with F ( D 4 ) = D 2 D 2 D 4 . Statement of the conjecture Specifically, Reduction to convenient polynomials Translation into prime values of polynomials � � �� � x + 1 � F x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . Shepherding the local factors a Sums of multiplicative functions Summary For deg F = 3, begin with F ( D 4 ) = D 3 D 3 D 6 .

  23. Friable values of Recursively use Schinzel’s construction polynomials Greg Martin D m : an unspecified polynomial of degree m Introduction Friable integers Friable values of polynomials Bounds for friable Example values of polynomials How friable can values of special polynomials be? deg F ( x ) = 4. Use Schinzel’s construction repeatedly: How friable can values of general polynomials be? Can we have lots of friable values? D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 Conjecture for prime D 84 = F ( D 21 ) = D 28 D 8 D 48 “score” = 16/7 values of polynomials Schinzel’s “Hypothesis H” D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 736/329 (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials For deg F = 2, begin with F ( D 4 ) = D 2 D 2 D 4 . Statement of the conjecture Specifically, Reduction to convenient polynomials Translation into prime values of polynomials � � �� � x + 1 � F x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . Shepherding the local factors a Sums of multiplicative functions Summary For deg F = 3, begin with F ( D 4 ) = D 3 D 3 D 6 .

  24. Friable values of Recursively use Schinzel’s construction polynomials Greg Martin D m : an unspecified polynomial of degree m Introduction Friable integers Friable values of polynomials Bounds for friable Example values of polynomials How friable can values of special polynomials be? deg F ( x ) = 4. Use Schinzel’s construction repeatedly: How friable can values of general polynomials be? Can we have lots of friable values? D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 Conjecture for prime D 84 = F ( D 21 ) = D 28 D 8 D 48 “score” = 16/7 values of polynomials Schinzel’s “Hypothesis H” D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 736/329 (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials For deg F = 2, begin with F ( D 4 ) = D 2 D 2 D 4 . Statement of the conjecture Specifically, Reduction to convenient polynomials Translation into prime values of polynomials � � �� � x + 1 � F x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . Shepherding the local factors a Sums of multiplicative functions Summary For deg F = 3, begin with F ( D 4 ) = D 3 D 3 D 6 .

  25. Friable values of Recursively use Schinzel’s construction polynomials Greg Martin D m : an unspecified polynomial of degree m Introduction Friable integers Friable values of polynomials Bounds for friable Example values of polynomials How friable can values of special polynomials be? deg F ( x ) = 4. Use Schinzel’s construction repeatedly: How friable can values of general polynomials be? Can we have lots of friable values? D 12 = F ( D 3 ) = D 4 D 8 “score” = 8/3 Conjecture for prime D 84 = F ( D 21 ) = D 28 D 8 D 48 “score” = 16/7 values of polynomials Schinzel’s “Hypothesis H” D 3984 = F ( D 987 ) = D 1316 D 376 D 48 D 2208 “score” = 736/329 (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials For deg F = 2, begin with F ( D 4 ) = D 2 D 2 D 4 . Statement of the conjecture Specifically, Reduction to convenient polynomials Translation into prime values of polynomials � � �� � x + 1 � F x + F ( x ) + F x + F ( x ) = F ( x ) · aF · D 4 . Shepherding the local factors a Sums of multiplicative functions Summary For deg F = 3, begin with F ( D 4 ) = D 3 D 3 D 6 .

  26. Friable values of How friable can values of general polynomials polynomials be? Greg Martin Introduction ∞ � � Friable integers 1 � d ≥ 4: define s ( d ) = d 1 − , where Friable values of polynomials u j ( d ) Bounds for friable j = 1 values of polynomials u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 How friable can values of special polynomials be? How friable can values of general s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 polynomials be? Can we have lots of friable values? Conjecture for prime values of polynomials Theorem Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , Conjecture for friable there are infinitely many numbers n for which F ( n ) is values of polynomials n s ( d ) -friable. Statement of the conjecture Reduction to convenient polynomials Translation into prime values of can be n ? -friable can be n ? -friable F ( n ) F ( n ) polynomials Shepherding the local factors degree 1 ε degree 5 3.46410 Sums of multiplicative functions degree 2 0.55902 degree 6 4.58258 Summary degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d

  27. Friable values of How friable can values of general polynomials polynomials be? Greg Martin Introduction ∞ � � Friable integers 1 � d ≥ 4: define s ( d ) = d 1 − , where Friable values of polynomials u j ( d ) Bounds for friable j = 1 values of polynomials u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 How friable can values of special polynomials be? How friable can values of general s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 polynomials be? Can we have lots of friable values? Conjecture for prime values of polynomials Theorem Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , Conjecture for friable there are infinitely many numbers n for which F ( n ) is values of polynomials n s ( d ) -friable. Statement of the conjecture Reduction to convenient polynomials Translation into prime values of can be n ? -friable can be n ? -friable F ( n ) F ( n ) polynomials Shepherding the local factors degree 1 ε degree 5 3.46410 Sums of multiplicative functions degree 2 0.55902 degree 6 4.58258 Summary degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d

  28. Friable values of How friable can values of general polynomials polynomials be? Greg Martin Introduction ∞ � � Friable integers 1 � d ≥ 4: define s ( d ) = d 1 − , where Friable values of polynomials u j ( d ) Bounds for friable j = 1 values of polynomials u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 How friable can values of special polynomials be? How friable can values of general s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 polynomials be? Can we have lots of friable values? Conjecture for prime values of polynomials Theorem Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , Conjecture for friable there are infinitely many numbers n for which F ( n ) is values of polynomials n s ( d ) -friable. Statement of the conjecture Reduction to convenient polynomials Translation into prime values of can be n ? -friable can be n ? -friable F ( n ) F ( n ) polynomials Shepherding the local factors degree 1 ε degree 5 3.46410 Sums of multiplicative functions degree 2 0.55902 degree 6 4.58258 Summary degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d

  29. Friable values of How friable can values of general polynomials polynomials be? Greg Martin Introduction ∞ � � Friable integers 1 � d ≥ 4: define s ( d ) = d 1 − , where Friable values of polynomials u j ( d ) Bounds for friable j = 1 values of polynomials u 1 ( d ) = d − 1 and u j + 1 ( d ) = u j ( d ) 2 − 2 How friable can values of special polynomials be? How friable can values of general s ( 2 ) = s ( 4 ) / 4 and s ( 3 ) = s ( 6 ) / 4 polynomials be? Can we have lots of friable values? Conjecture for prime values of polynomials Theorem Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H (Schinzel, 1967) Given a polynomial F ( x ) of degree d ≥ 2 , Conjecture for friable there are infinitely many numbers n for which F ( n ) is values of polynomials n s ( d ) -friable. Statement of the conjecture Reduction to convenient polynomials Translation into prime values of can be n ? -friable can be n ? -friable F ( n ) F ( n ) polynomials Shepherding the local factors degree 1 ε degree 5 3.46410 Sums of multiplicative functions degree 2 0.55902 degree 6 4.58258 Summary degree 3 1.14564 degree 7 5.65685 degree 4 2.23607 degree d ≈ d − 1 − 2 / d

  30. Friable values of Polynomial substitution yields small lower polynomials bounds Greg Martin Introduction Friable integers Special case Friable values of polynomials Bounds for friable Given a quadratic polynomial F ( x ) , there are infinitely many values of polynomials numbers n for which F ( n ) is n 0 . 55902 -friable. How friable can values of special polynomials be? How friable can values of general polynomials be? Can we have lots of friable values? Example Conjecture for prime values of polynomials To obtain n for which F ( n ) is n 0 . 56 -friable: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 4 / 7 > 0 . 56 Conjecture for friable D 7896 = F ( D 3948 ) “score” = 92 / 329 values of polynomials = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 Statement of the conjecture Reduction to convenient polynomials The counting function of such n is about x 1 / 3948 . Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions “Improvement” Balog, M., Wooley can get x 2 / 3948 and an Summary analogous improvement for deg F = 3.

  31. Friable values of Polynomial substitution yields small lower polynomials bounds Greg Martin Introduction Friable integers Special case Friable values of polynomials Bounds for friable Given a quadratic polynomial F ( x ) , there are infinitely many values of polynomials numbers n for which F ( n ) is n 0 . 55902 -friable. How friable can values of special polynomials be? How friable can values of general polynomials be? Can we have lots of friable values? Example Conjecture for prime values of polynomials To obtain n for which F ( n ) is n 0 . 56 -friable: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 4 / 7 > 0 . 56 Conjecture for friable D 7896 = F ( D 3948 ) “score” = 92 / 329 values of polynomials = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 Statement of the conjecture Reduction to convenient polynomials The counting function of such n is about x 1 / 3948 . Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions “Improvement” Balog, M., Wooley can get x 2 / 3948 and an Summary analogous improvement for deg F = 3.

  32. Friable values of Polynomial substitution yields small lower polynomials bounds Greg Martin Introduction Friable integers Special case Friable values of polynomials Bounds for friable Given a quadratic polynomial F ( x ) , there are infinitely many values of polynomials numbers n for which F ( n ) is n 0 . 55902 -friable. How friable can values of special polynomials be? How friable can values of general polynomials be? Can we have lots of friable values? Example Conjecture for prime values of polynomials To obtain n for which F ( n ) is n 0 . 56 -friable: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 4 / 7 > 0 . 56 Conjecture for friable D 7896 = F ( D 3948 ) “score” = 92 / 329 values of polynomials = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 Statement of the conjecture Reduction to convenient polynomials The counting function of such n is about x 1 / 3948 . Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions “Improvement” Balog, M., Wooley can get x 2 / 3948 and an Summary analogous improvement for deg F = 3.

  33. Friable values of Polynomial substitution yields small lower polynomials bounds Greg Martin Introduction Friable integers Special case Friable values of polynomials Bounds for friable Given a quadratic polynomial F ( x ) , there are infinitely many values of polynomials numbers n for which F ( n ) is n 0 . 55902 -friable. How friable can values of special polynomials be? How friable can values of general polynomials be? Can we have lots of friable values? Example Conjecture for prime values of polynomials To obtain n for which F ( n ) is n 0 . 56 -friable: Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H D 168 = F ( D 84 ) = D 42 D 42 D 28 D 8 D 48 “score” = 4 / 7 > 0 . 56 Conjecture for friable D 7896 = F ( D 3948 ) “score” = 92 / 329 values of polynomials = D 1974 D 1974 D 1316 D 376 D 48 D 2208 < 0 . 56 Statement of the conjecture Reduction to convenient polynomials The counting function of such n is about x 1 / 3948 . Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions “Improvement” Balog, M., Wooley can get x 2 / 3948 and an Summary analogous improvement for deg F = 3.

  34. Friable values of Can we have lots of friable values? polynomials Greg Martin Introduction Our expectation Friable integers Friable values of polynomials For any ε > 0, a positive proportion of values F ( n ) are Bounds for friable n ε -friable. values of polynomials How friable can values of special polynomials be? We know this for: How friable can values of general polynomials be? Can we have lots of friable values? linear polynomials (arithmetic progressions) Conjecture for prime values of polynomials Hildebrand, then Balog and Ruzsa: F ( n ) = n ( an + b ) , Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) values n ε -friable for any ε > 0 A uniform version of Hypothesis H Conjecture for friable Hildebrand: F ( n ) = ( n + 1 ) · · · ( n + L ) , values values of polynomials n β -friable for any β > e − 1 / ( L − 1 ) Statement of the conjecture Reduction to convenient polynomials L , so β > e − 1 / L is trivial Note: ρ ( e − 1 / L ) = 1 − 1 Translation into prime values of polynomials Shepherding the local factors Dartyge: F ( n ) = n 2 + 1, values n β -friable for any Sums of multiplicative functions β > 149 / 179 Summary

  35. Friable values of Can we have lots of friable values? polynomials Greg Martin Introduction Our expectation Friable integers Friable values of polynomials For any ε > 0, a positive proportion of values F ( n ) are Bounds for friable n ε -friable. values of polynomials How friable can values of special polynomials be? We know this for: How friable can values of general polynomials be? Can we have lots of friable values? linear polynomials (arithmetic progressions) Conjecture for prime values of polynomials Hildebrand, then Balog and Ruzsa: F ( n ) = n ( an + b ) , Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) values n ε -friable for any ε > 0 A uniform version of Hypothesis H Conjecture for friable Hildebrand: F ( n ) = ( n + 1 ) · · · ( n + L ) , values values of polynomials n β -friable for any β > e − 1 / ( L − 1 ) Statement of the conjecture Reduction to convenient polynomials L , so β > e − 1 / L is trivial Note: ρ ( e − 1 / L ) = 1 − 1 Translation into prime values of polynomials Shepherding the local factors Dartyge: F ( n ) = n 2 + 1, values n β -friable for any Sums of multiplicative functions β > 149 / 179 Summary

  36. Friable values of Can we have lots of friable values? polynomials Greg Martin Introduction Our expectation Friable integers Friable values of polynomials For any ε > 0, a positive proportion of values F ( n ) are Bounds for friable n ε -friable. values of polynomials How friable can values of special polynomials be? We know this for: How friable can values of general polynomials be? Can we have lots of friable values? linear polynomials (arithmetic progressions) Conjecture for prime values of polynomials Hildebrand, then Balog and Ruzsa: F ( n ) = n ( an + b ) , Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) values n ε -friable for any ε > 0 A uniform version of Hypothesis H Conjecture for friable Hildebrand: F ( n ) = ( n + 1 ) · · · ( n + L ) , values values of polynomials n β -friable for any β > e − 1 / ( L − 1 ) Statement of the conjecture Reduction to convenient polynomials L , so β > e − 1 / L is trivial Note: ρ ( e − 1 / L ) = 1 − 1 Translation into prime values of polynomials Shepherding the local factors Dartyge: F ( n ) = n 2 + 1, values n β -friable for any Sums of multiplicative functions β > 149 / 179 Summary

  37. Friable values of polynomials Greg Martin Theorem (Dartyge, M., Tenenbaum, 2001) Introduction Let F ( x ) be any polynomial, let d be the highest degree of Friable integers Friable values of polynomials any irreducible factor of F, and let F have exactly K distinct Bounds for friable irreducible factors of degree d. Then for any ε > 0 , a values of polynomials positive proportion of values F ( n ) are n d − 1 / K + ε -friable. How friable can values of special polynomials be? How friable can values of general polynomials be? Remark: for friability of level n d − 1 or higher, only irreducible Can we have lots of friable values? Conjecture for prime factors of degree ≥ d matter values of polynomials Schinzel’s “Hypothesis H” Trivial: n d -friable (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Can remove the ε at the cost of the counting function: the values of polynomials number of n ≤ x for which F ( n ) is n d − 1 / K -friable is Statement of the conjecture Reduction to convenient polynomials Translation into prime values of x polynomials ≫ ( log x ) K ( log 4 − 1 + ε ) . Shepherding the local factors Sums of multiplicative functions Summary

  38. Friable values of polynomials Greg Martin Theorem (Dartyge, M., Tenenbaum, 2001) Introduction Let F ( x ) be any polynomial, let d be the highest degree of Friable integers Friable values of polynomials any irreducible factor of F, and let F have exactly K distinct Bounds for friable irreducible factors of degree d. Then for any ε > 0 , a values of polynomials positive proportion of values F ( n ) are n d − 1 / K + ε -friable. How friable can values of special polynomials be? How friable can values of general polynomials be? Remark: for friability of level n d − 1 or higher, only irreducible Can we have lots of friable values? Conjecture for prime factors of degree ≥ d matter values of polynomials Schinzel’s “Hypothesis H” Trivial: n d -friable (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Can remove the ε at the cost of the counting function: the values of polynomials number of n ≤ x for which F ( n ) is n d − 1 / K -friable is Statement of the conjecture Reduction to convenient polynomials Translation into prime values of x polynomials ≫ ( log x ) K ( log 4 − 1 + ε ) . Shepherding the local factors Sums of multiplicative functions Summary

  39. Friable values of polynomials Greg Martin Introduction 1 Introduction Friable integers Friable values of polynomials Bounds for friable values of polynomials 2 Bounds for friable values of polynomials How friable can values of special polynomials be? How friable can values of general polynomials be? 3 Conjecture for prime values of polynomials Can we have lots of friable values? Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Conjecture for prime values of polynomials A uniform version of Hypothesis H Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H 4 Conjecture for friable values of polynomials Conjecture for friable values of polynomials Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  40. Friable values of Schinzel’s “Hypothesis H” (Bateman–Horn polynomials conjecture) Greg Martin Introduction Definition Friable integers Friable values of polynomials π ( F ; x ) = # { n ≤ x : Bounds for friable values of polynomials f ( n ) is prime for each irreducible factor f of F } How friable can values of special polynomials be? How friable can values of general polynomials be? Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: Can we have lots of friable values? Conjecture for prime values of polynomials � dt Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) li ( F ; x ) = log | F 1 ( t ) | . . . log | F L ( t ) | . A uniform version of Hypothesis H Conjecture for friable 0 < t < x values of polynomials min {| F 1 ( t ) | ,..., | F L ( t ) |}≥ 2 Statement of the conjecture � − L � Reduction to convenient � � 1 − 1 1 − σ ( F ; p ) polynomials � H ( F ) = . Translation into prime values of p p polynomials p Shepherding the local factors Sums of multiplicative functions L : the number of distinct irreducible factors of F Summary σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n )

  41. Friable values of Schinzel’s “Hypothesis H” (Bateman–Horn polynomials conjecture) Greg Martin Introduction Definition Friable integers Friable values of polynomials π ( F ; x ) = # { n ≤ x : Bounds for friable values of polynomials f ( n ) is prime for each irreducible factor f of F } How friable can values of special polynomials be? How friable can values of general polynomials be? Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: Can we have lots of friable values? Conjecture for prime values of polynomials � dt Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) li ( F ; x ) = log | F 1 ( t ) | . . . log | F L ( t ) | . A uniform version of Hypothesis H Conjecture for friable 0 < t < x values of polynomials min {| F 1 ( t ) | ,..., | F L ( t ) |}≥ 2 Statement of the conjecture � − L � Reduction to convenient � � 1 − 1 1 − σ ( F ; p ) polynomials � H ( F ) = . Translation into prime values of p p polynomials p Shepherding the local factors Sums of multiplicative functions L : the number of distinct irreducible factors of F Summary σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n )

  42. Friable values of Schinzel’s “Hypothesis H” (Bateman–Horn polynomials conjecture) Greg Martin Introduction Definition Friable integers Friable values of polynomials π ( F ; x ) = # { n ≤ x : Bounds for friable values of polynomials f ( n ) is prime for each irreducible factor f of F } How friable can values of special polynomials be? How friable can values of general polynomials be? Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: Can we have lots of friable values? Conjecture for prime values of polynomials � dt Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) li ( F ; x ) = log | F 1 ( t ) | . . . log | F L ( t ) | . A uniform version of Hypothesis H Conjecture for friable 0 < t < x values of polynomials min {| F 1 ( t ) | ,..., | F L ( t ) |}≥ 2 Statement of the conjecture � − L � Reduction to convenient � � 1 − 1 1 − σ ( F ; p ) polynomials � H ( F ) = . Translation into prime values of p p polynomials p Shepherding the local factors Sums of multiplicative functions L : the number of distinct irreducible factors of F Summary σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n )

  43. Friable values of Schinzel’s “Hypothesis H” (Bateman–Horn polynomials conjecture) Greg Martin Introduction Definition Friable integers Friable values of polynomials π ( F ; x ) = # { n ≤ x : Bounds for friable values of polynomials f ( n ) is prime for each irreducible factor f of F } How friable can values of special polynomials be? How friable can values of general polynomials be? Conjecture: π ( F ; x ) is asymptotic to H ( F ) li ( F ; x ) , where: Can we have lots of friable values? Conjecture for prime values of polynomials � dt Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) li ( F ; x ) = log | F 1 ( t ) | . . . log | F L ( t ) | . A uniform version of Hypothesis H Conjecture for friable 0 < t < x values of polynomials min {| F 1 ( t ) | ,..., | F L ( t ) |}≥ 2 Statement of the conjecture � − L � Reduction to convenient � � 1 − 1 1 − σ ( F ; p ) polynomials � H ( F ) = . Translation into prime values of p p polynomials p Shepherding the local factors Sums of multiplicative functions L : the number of distinct irreducible factors of F Summary σ ( F ; n ) : the number of solutions of F ( a ) ≡ 0 (mod n )

  44. Friable values of A uniform version of Hypothesis H polynomials Greg Martin Hypothesis UH Introduction Friable integers H ( F ) t Friable values of polynomials π ( F ; t ) − H ( F ) li ( F ; t ) ≪ d , B 1 + Bounds for friable ( log t ) L + 1 values of polynomials How friable can values of special uniformly for all polynomials F of degree d with L distinct polynomials be? How friable can values of general irreducible factors, each of which has coefficients bounded polynomials be? Can we have lots of friable values? by t B in absolute value. Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” t (Bateman–Horn conjecture) li ( F ; t ) is asymptotic to ( log t ) L for fixed F A uniform version of Hypothesis H Conjecture for friable values of polynomials For d = K = 1, equivalent to expected number of Statement of the conjecture primes, in an interval of length y = x ε near x , in an Reduction to convenient polynomials Translation into prime values of arithmetic progression to a modulus q ≤ y 1 − ε polynomials Shepherding the local factors Sums of multiplicative functions Don’t really need this strong a uniformity, but rather on Summary average over some funny family to be described later

  45. Friable values of A uniform version of Hypothesis H polynomials Greg Martin Hypothesis UH Introduction Friable integers H ( F ) t Friable values of polynomials π ( F ; t ) − H ( F ) li ( F ; t ) ≪ d , B 1 + Bounds for friable ( log t ) L + 1 values of polynomials How friable can values of special uniformly for all polynomials F of degree d with L distinct polynomials be? How friable can values of general irreducible factors, each of which has coefficients bounded polynomials be? Can we have lots of friable values? by t B in absolute value. Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” t (Bateman–Horn conjecture) li ( F ; t ) is asymptotic to ( log t ) L for fixed F A uniform version of Hypothesis H Conjecture for friable values of polynomials For d = K = 1, equivalent to expected number of Statement of the conjecture primes, in an interval of length y = x ε near x , in an Reduction to convenient polynomials Translation into prime values of arithmetic progression to a modulus q ≤ y 1 − ε polynomials Shepherding the local factors Sums of multiplicative functions Don’t really need this strong a uniformity, but rather on Summary average over some funny family to be described later

  46. Friable values of polynomials Greg Martin 1 Introduction Introduction Friable integers Friable values of polynomials 2 Bounds for friable values of polynomials Bounds for friable values of polynomials How friable can values of special polynomials be? Conjecture for prime values of polynomials 3 How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Conjecture for friable values of polynomials 4 values of polynomials Statement of the conjecture Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) Reduction to convenient polynomials A uniform version of Hypothesis H Translation into prime values of polynomials Conjecture for friable values of polynomials Shepherding the local factors Statement of the conjecture Sums of multiplicative functions Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  47. Friable values of What would we expect on probablistic polynomials grounds? Greg Martin Introduction Friable integers Friable values of polynomials Let F ( x ) = f 1 ( x ) · · · f L ( x ) , where deg f j ( x ) = d j . Let u > 0. Bounds for friable values of polynomials f j ( n ) is roughly n d j , and integers of that size are How friable can values of special polynomials be? n 1 / u -friable with probability ρ ( d j u ) . How friable can values of general polynomials be? Can we have lots of friable values? Are the friabilities of the various factors f j ( n ) Conjecture for prime independent? This would lead to a prediction involving values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) L A uniform version of Hypothesis H � ρ ( d j u ) . x Conjecture for friable values of polynomials j = 1 Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials What about local densities depending on the arithmetic Shepherding the local factors Sums of multiplicative functions of F (as in Hypothesis H)? Summary

  48. Friable values of What would we expect on probablistic polynomials grounds? Greg Martin Introduction Friable integers Friable values of polynomials Let F ( x ) = f 1 ( x ) · · · f L ( x ) , where deg f j ( x ) = d j . Let u > 0. Bounds for friable values of polynomials f j ( n ) is roughly n d j , and integers of that size are How friable can values of special polynomials be? n 1 / u -friable with probability ρ ( d j u ) . How friable can values of general polynomials be? Can we have lots of friable values? Are the friabilities of the various factors f j ( n ) Conjecture for prime independent? This would lead to a prediction involving values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) L A uniform version of Hypothesis H � ρ ( d j u ) . x Conjecture for friable values of polynomials j = 1 Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials What about local densities depending on the arithmetic Shepherding the local factors Sums of multiplicative functions of F (as in Hypothesis H)? Summary

  49. Friable values of What would we expect on probablistic polynomials grounds? Greg Martin Introduction Friable integers Friable values of polynomials Let F ( x ) = f 1 ( x ) · · · f L ( x ) , where deg f j ( x ) = d j . Let u > 0. Bounds for friable values of polynomials f j ( n ) is roughly n d j , and integers of that size are How friable can values of special polynomials be? n 1 / u -friable with probability ρ ( d j u ) . How friable can values of general polynomials be? Can we have lots of friable values? Are the friabilities of the various factors f j ( n ) Conjecture for prime independent? This would lead to a prediction involving values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) L A uniform version of Hypothesis H � ρ ( d j u ) . x Conjecture for friable values of polynomials j = 1 Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials What about local densities depending on the arithmetic Shepherding the local factors Sums of multiplicative functions of F (as in Hypothesis H)? Summary

  50. Friable values of Conjecture for friable values of polynomials polynomials Greg Martin Conjecture Introduction Friable integers Let F ( x ) be any polynomial, let f 1 , . . . , f L be its distinct Friable values of polynomials irreducible factors, and let d 1 , . . . , d L be their degrees. Then Bounds for friable values of polynomials How friable can values of special L � x � polynomials be? � Ψ( F ; x , x 1 / u ) = x ρ ( d j u ) + O How friable can values of general polynomials be? log x Can we have lots of friable values? j = 1 Conjecture for prime values of polynomials for all 0 < u . Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials Statement of the conjecture If F irreducible: Ψ( F ; x , x 1 / u ) = x ρ ( du ) + O ( x / log x ) for Reduction to convenient polynomials 0 < u . Translation into prime values of polynomials Shepherding the local factors Remark: Rather more controversial than Sums of multiplicative functions Hypothesis H. Summary

  51. Friable values of Conjecture for friable values of polynomials polynomials Greg Martin Theorem (M., 2002) Introduction Friable integers Assume Hypothesis UH. Let F ( x ) be any polynomial, let Friable values of polynomials f 1 , . . . , f L be its distinct irreducible factors, and let d 1 , . . . , d L Bounds for friable values of polynomials be their degrees. Let d = max { d 1 , . . . , d L } , and let F have How friable can values of special exactly K distinct irreducible factors of degree d. Then polynomials be? How friable can values of general polynomials be? Can we have lots of friable values? L � x � � Ψ( F ; x , x 1 / u ) = x Conjecture for prime ρ ( d j u ) + O values of polynomials log x j = 1 Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H for all 0 < u < 1 / ( d − 1 / K ) . Conjecture for friable values of polynomials Statement of the conjecture If F irreducible: Ψ( F ; x , x 1 / u ) = x ρ ( du ) + O ( x / log x ) for Reduction to convenient polynomials 0 < u < 1 / ( d − 1 ) . Translation into prime values of polynomials Shepherding the local factors Trivial: 0 < u < 1 / d . Sums of multiplicative functions Summary Reason to talk about more general K : There is one part of the argument that causes an additional difficulty when K > 1.

  52. Friable values of Conjecture for friable values of polynomials polynomials Greg Martin Theorem (M., 2002) Introduction Friable integers Assume Hypothesis UH. Let F ( x ) be any polynomial, let Friable values of polynomials f 1 , . . . , f L be its distinct irreducible factors, and let d 1 , . . . , d L Bounds for friable values of polynomials be their degrees. Let d = max { d 1 , . . . , d L } , and let F have How friable can values of special exactly K distinct irreducible factors of degree d. Then polynomials be? How friable can values of general polynomials be? Can we have lots of friable values? L � x � � Ψ( F ; x , x 1 / u ) = x Conjecture for prime ρ ( d j u ) + O values of polynomials log x j = 1 Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H for all 0 < u < 1 / ( d − 1 / K ) . Conjecture for friable values of polynomials Statement of the conjecture If F irreducible: Ψ( F ; x , x 1 / u ) = x ρ ( du ) + O ( x / log x ) for Reduction to convenient polynomials 0 < u < 1 / ( d − 1 ) . Translation into prime values of polynomials Shepherding the local factors Trivial: 0 < u < 1 / d . Sums of multiplicative functions Summary Reason to talk about more general K : There is one part of the argument that causes an additional difficulty when K > 1.

  53. Friable values of Reduction to convenient polynomials polynomials Greg Martin Without loss of generality, we may assume: Introduction Friable integers F ( x ) is the product of distinct irreducible polynomials 1 Friable values of polynomials f 1 ( x ) , . . . , f K ( x ) , all of the same degree d . Bounds for friable values of polynomials F ( x ) takes at least one nonzero value modulo every 2 How friable can values of special polynomials be? prime. How friable can values of general polynomials be? Can we have lots of friable values? No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) 3 Conjecture for prime have a common zero modulo any prime. values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H (1) is acceptable since the friability level exceeds x d − 1 . Conjecture for friable values of polynomials Statement of the conjecture (2) is not a necessary condition to have friable values Reduction to convenient polynomials (as it is to have prime values). Nevertheless, we can Translation into prime values of polynomials still reduce to this case. Shepherding the local factors Sums of multiplicative functions Both (2) and (3) are achieved by looking at values of Summary F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately.

  54. Friable values of Reduction to convenient polynomials polynomials Greg Martin Without loss of generality, we may assume: Introduction Friable integers F ( x ) is the product of distinct irreducible polynomials 1 Friable values of polynomials f 1 ( x ) , . . . , f K ( x ) , all of the same degree d . Bounds for friable values of polynomials F ( x ) takes at least one nonzero value modulo every 2 How friable can values of special polynomials be? prime. How friable can values of general polynomials be? Can we have lots of friable values? No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) 3 Conjecture for prime have a common zero modulo any prime. values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H (1) is acceptable since the friability level exceeds x d − 1 . Conjecture for friable values of polynomials Statement of the conjecture (2) is not a necessary condition to have friable values Reduction to convenient polynomials (as it is to have prime values). Nevertheless, we can Translation into prime values of polynomials still reduce to this case. Shepherding the local factors Sums of multiplicative functions Both (2) and (3) are achieved by looking at values of Summary F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately.

  55. Friable values of Reduction to convenient polynomials polynomials Greg Martin Without loss of generality, we may assume: Introduction Friable integers F ( x ) is the product of distinct irreducible polynomials 1 Friable values of polynomials f 1 ( x ) , . . . , f K ( x ) , all of the same degree d . Bounds for friable values of polynomials F ( x ) takes at least one nonzero value modulo every 2 How friable can values of special polynomials be? prime. How friable can values of general polynomials be? Can we have lots of friable values? No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) 3 Conjecture for prime have a common zero modulo any prime. values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H (1) is acceptable since the friability level exceeds x d − 1 . Conjecture for friable values of polynomials Statement of the conjecture (2) is not a necessary condition to have friable values Reduction to convenient polynomials (as it is to have prime values). Nevertheless, we can Translation into prime values of polynomials still reduce to this case. Shepherding the local factors Sums of multiplicative functions Both (2) and (3) are achieved by looking at values of Summary F ( x ) on suitable arithmetic progressions F ( Qx + R ) separately.

  56. Friable values of Reduction to convenient polynomials polynomials Greg Martin Without loss of generality, we may assume: Introduction Friable integers F ( x ) is the product of distinct irreducible polynomials 1 Friable values of polynomials f 1 ( x ) , . . . , f K ( x ) , all of the same degree d . Bounds for friable values of polynomials F ( x ) takes at least one nonzero value modulo every 2 How friable can values of special polynomials be? prime. How friable can values of general polynomials be? Can we have lots of friable values? No two distinct irreducible factors f i ( x ) , f j ( x ) of F ( x ) 3 Conjecture for prime have a common zero modulo any prime. values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Under (1), we want to prove that values of polynomials Statement of the conjecture Reduction to convenient � x � polynomials Ψ( F ; x , x 1 / u ) = x ρ ( du ) K + O Translation into prime values of polynomials log x Shepherding the local factors Sums of multiplicative functions for all 0 < u < 1 / ( d − 1 / K ) . Summary

  57. Friable values of Inclusion-exclusion on irreducible factors polynomials Greg Martin Proposition Introduction Friable integers Let F be a primitive polynomial, and let F 1 , . . . , F K denote Friable values of polynomials the distinct irreducible factors of F. Then for x ≥ y ≥ 1 , Bounds for friable values of polynomials How friable can values of special � ( − 1 ) k � Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . polynomials be? How friable can values of general polynomials be? 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” Definition (Bateman–Horn conjecture) A uniform version of Hypothesis H M ( f ; x , y ) = # { 1 ≤ n ≤ x : for each irreducible factor g of f , Conjecture for friable values of polynomials there exists a prime p > y such that p | g ( n ) } . Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  58. Friable values of Inclusion-exclusion on irreducible factors polynomials Greg Martin Proposition Introduction Friable integers Let F be a primitive polynomial, and let F 1 , . . . , F K denote Friable values of polynomials the distinct irreducible factors of F. Then for x ≥ y ≥ 1 , Bounds for friable values of polynomials How friable can values of special � ( − 1 ) k � Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . polynomials be? How friable can values of general polynomials be? 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” Definition (Bateman–Horn conjecture) A uniform version of Hypothesis H M ( f ; x , y ) = # { 1 ≤ n ≤ x : for each irreducible factor g of f , Conjecture for friable values of polynomials there exists a prime p > y such that p | g ( n ) } . Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  59. Friable values of Inclusion-exclusion on irreducible factors polynomials Greg Martin Proposition Introduction Friable integers Let F be a primitive polynomial, and let F 1 , . . . , F K denote Friable values of polynomials the distinct irreducible factors of F. Then for x ≥ y ≥ 1 , Bounds for friable values of polynomials How friable can values of special � ( − 1 ) k � Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . polynomials be? How friable can values of general polynomials be? 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) If we knew that M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k , then A uniform version of Hypothesis H Conjecture for friable Ψ( F ; x , x 1 / u ) ∼ x + � ( − 1 ) k � x ( log du ) k values of polynomials Statement of the conjecture Reduction to convenient 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K polynomials � � Translation into prime values of � � K ( − log du ) k polynomials � = x 1 + Shepherding the local factors k Sums of multiplicative functions 1 ≤ k ≤ K Summary = x ( 1 − log du ) K = x ρ ( du ) K .

  60. Friable values of Inclusion-exclusion on irreducible factors polynomials Greg Martin Proposition Introduction Friable integers Let F be a primitive polynomial, and let F 1 , . . . , F K denote Friable values of polynomials the distinct irreducible factors of F. Then for x ≥ y ≥ 1 , Bounds for friable values of polynomials How friable can values of special � ( − 1 ) k � Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . polynomials be? How friable can values of general polynomials be? 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) If we knew that M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k , then A uniform version of Hypothesis H Conjecture for friable Ψ( F ; x , x 1 / u ) ∼ x + � ( − 1 ) k � x ( log du ) k values of polynomials Statement of the conjecture Reduction to convenient 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K polynomials � � Translation into prime values of � � K ( − log du ) k polynomials � = x 1 + Shepherding the local factors k Sums of multiplicative functions 1 ≤ k ≤ K Summary = x ( 1 − log du ) K = x ρ ( du ) K .

  61. Friable values of Inclusion-exclusion on irreducible factors polynomials Greg Martin Proposition Introduction Friable integers Let F be a primitive polynomial, and let F 1 , . . . , F K denote Friable values of polynomials the distinct irreducible factors of F. Then for x ≥ y ≥ 1 , Bounds for friable values of polynomials How friable can values of special � ( − 1 ) k � Ψ( F ; x , y ) = ⌊ x ⌋ + M ( F i 1 . . . F i k ; x , y ) . polynomials be? How friable can values of general polynomials be? 1 ≤ k ≤ K 1 ≤ i 1 < ··· < i k ≤ K Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” Definition (Bateman–Horn conjecture) A uniform version of Hypothesis H M ( f ; x , y ) = # { 1 ≤ n ≤ x : for each irreducible factor g of f , Conjecture for friable values of polynomials there exists a prime p > y such that p | g ( n ) } . Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials We want to prove M ( F i 1 . . . F i k ; x , x 1 / u ) ∼ x ( log du ) k . To do Shepherding the local factors Sums of multiplicative functions this, we sort by the values n j = F i j ( n ) / p j , among those n Summary counted by M ( F i 1 . . . F i k ; x , x 1 / u ) .

  62. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  63. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable values of polynomials DON’T PANIC Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  64. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H not important Conjecture for friable values of polynomials Statement of the conjecture ξ j = f j ( x ) ≈ x d Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  65. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H not important Conjecture for friable values of polynomials Statement of the conjecture η n 1 ,..., n k ≈ ( y max { n 1 , . . . , n k } ) 1 / d ( n 1 · · · n k ) − 1 Reduction to convenient polynomials Translation into prime values of polynomials It’s here only because the large primes dividing f j ( n ) had to Shepherding the local factors Sums of multiplicative functions exceed y . (Later we’ll take y = x 1 / u .) Summary

  66. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H fairly important Conjecture for friable values of polynomials Statement of the conjecture � R ( f ; n 1 , . . . , n k ) = b (mod n 1 · · · n k ) : Reduction to convenient polynomials � n 1 | f 1 ( b ) , n 2 | f 2 ( b ) , . . . , n k | f k ( b ) Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  67. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H rather important Conjecture for friable values of polynomials Statement of the conjecture f n 1 ··· n k , b ( t ) = f ( n 1 · · · n k t + b ) Reduction to convenient ∈ Z [ x ] polynomials n 1 · · · n k Translation into prime values of polynomials Shepherding the local factors In fact, a good understanding of the family f n 1 ··· n k , b is Sums of multiplicative functions Summary necessary even to treat error terms. However, we’ll only include the details when treating the main term.

  68. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H rather important Conjecture for friable values of polynomials Statement of the conjecture f n 1 ··· n k , b ( t ) = f ( n 1 · · · n k t + b ) Reduction to convenient ∈ Z [ x ] polynomials n 1 · · · n k Translation into prime values of polynomials Shepherding the local factors In fact, a good understanding of the family f n 1 ··· n k , b is Sums of multiplicative functions Summary necessary even to treat error terms. However, we’ll only include the details when treating the main term.

  69. Friable values of polynomials Proposition Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � M ( f ; x , y ) = · · · Friable values of polynomials Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? � x − b � How friable can values of general � � π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) . polynomials be? n 1 · · · n k Can we have lots of friable values? Conjecture for prime values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H First: concentrate on Conjecture for friable values of polynomials Statement of the conjecture x − b � � Reduction to convenient π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) polynomials n 1 · · · n k Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Summary

  70. Friable values of Understanding M ( f ; x , y ) inside out polynomials Greg Martin x − b � � Look at π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) n 1 · · · n k Introduction Friable integers Upper bound sieve (Brun, Selberg): Friable values of polynomials Bounds for friable values of polynomials � x − b � � H ( f n 1 ··· n k , b ) x / n 1 · · · n k � π f n 1 ··· n k , b ; + O How friable can values of special polynomials be? ( log x ) k + 1 n 1 · · · n k How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Main term for π ( f ; x ) (we use Hypothesis UH here!): values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) � x − b � � H ( f n 1 ··· n k , b ) x � A uniform version of Hypothesis H H ( f n 1 ··· n k , b ) li f n 1 ··· n k , b ; + O Conjecture for friable n 1 · · · n k ( log x ) k + 1 n 1 · · · n k values of polynomials Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions li is a pretty smooth function: Summary H ( f n 1 ··· n k , b ) x / n 1 · · · n k � H ( f n 1 ··· n k , b ) x � log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) + O n 1 · · · n k ( log x ) k + 1

  71. Friable values of Understanding M ( f ; x , y ) inside out polynomials Greg Martin x − b � � Look at π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) n 1 · · · n k Introduction Friable integers Upper bound sieve (Brun, Selberg): Friable values of polynomials Bounds for friable values of polynomials � x − b � � H ( f n 1 ··· n k , b ) x / n 1 · · · n k � π f n 1 ··· n k , b ; + O How friable can values of special polynomials be? ( log x ) k + 1 n 1 · · · n k How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Main term for π ( f ; x ) (we use Hypothesis UH here!): values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) � x − b � � H ( f n 1 ··· n k , b ) x � A uniform version of Hypothesis H H ( f n 1 ··· n k , b ) li f n 1 ··· n k , b ; + O Conjecture for friable n 1 · · · n k ( log x ) k + 1 n 1 · · · n k values of polynomials Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions li is a pretty smooth function: Summary H ( f n 1 ··· n k , b ) x / n 1 · · · n k � H ( f n 1 ··· n k , b ) x � log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) + O n 1 · · · n k ( log x ) k + 1

  72. Friable values of Understanding M ( f ; x , y ) inside out polynomials Greg Martin x − b � � Look at π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) n 1 · · · n k Introduction Friable integers Upper bound sieve (Brun, Selberg): Friable values of polynomials Bounds for friable values of polynomials � x − b � � H ( f n 1 ··· n k , b ) x / n 1 · · · n k � π f n 1 ··· n k , b ; + O How friable can values of special polynomials be? ( log x ) k + 1 n 1 · · · n k How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Main term for π ( f ; x ) (we use Hypothesis UH here!): values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) � x − b � � H ( f n 1 ··· n k , b ) x � A uniform version of Hypothesis H H ( f n 1 ··· n k , b ) li f n 1 ··· n k , b ; + O Conjecture for friable n 1 · · · n k ( log x ) k + 1 n 1 · · · n k values of polynomials Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions li is a pretty smooth function: Summary H ( f n 1 ··· n k , b ) x / n 1 · · · n k � H ( f n 1 ··· n k , b ) x � log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) + O n 1 · · · n k ( log x ) k + 1

  73. Friable values of Understanding M ( f ; x , y ) inside out polynomials Greg Martin x − b � � Look at π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) n 1 · · · n k Introduction Friable integers Upper bound sieve (Brun, Selberg): Friable values of polynomials Bounds for friable values of polynomials � x − b � � H ( f n 1 ··· n k , b ) x / n 1 · · · n k � π f n 1 ··· n k , b ; + O How friable can values of special polynomials be? ( log x ) k + 1 n 1 · · · n k How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Main term for π ( f ; x ) (we use Hypothesis UH here!): values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) � x − b � � H ( f n 1 ··· n k , b ) x � A uniform version of Hypothesis H H ( f n 1 ··· n k , b ) li f n 1 ··· n k , b ; + O Conjecture for friable n 1 · · · n k ( log x ) k + 1 n 1 · · · n k values of polynomials Statement of the conjecture Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions li is a pretty smooth function: Summary H ( f n 1 ··· n k , b ) x / n 1 · · · n k � H ( f n 1 ··· n k , b ) x � log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) + O n 1 · · · n k ( log x ) k + 1

  74. Friable values of Understanding M ( f ; x , y ) inside out polynomials Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � Friable values of polynomials M ( f ; x , y ) = · · · Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? How friable can values of general � x − b � � � polynomials be? π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) Can we have lots of friable values? n 1 · · · n k Conjecture for prime � � values of polynomials � � � = · · · H ( f n 1 ··· n k , b ) Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) A uniform version of Hypothesis H ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Conjecture for friable values of polynomials � �� x / n 1 · · · n k 1 � Statement of the conjecture × 1 + O . Reduction to convenient log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) log x polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions Now we have: Summary H ( f n 1 ··· n k , b ) x / n 1 · · · n k � H ( f n 1 ··· n k , b ) x � log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) + O n 1 · · · n k ( log x ) k + 1

  75. Friable values of Understanding M ( f ; x , y ) inside out polynomials Greg Martin For f = f 1 . . . f k and x and y sufficiently large, Introduction Friable integers � � � Friable values of polynomials M ( f ; x , y ) = · · · Bounds for friable n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) How friable can values of special polynomials be? How friable can values of general � x − b � � � polynomials be? π f n 1 ··· n k , b ; − π ( f n 1 ··· n k , b ; η n 1 ,..., n k ) Can we have lots of friable values? n 1 · · · n k Conjecture for prime � � values of polynomials � � � = · · · H ( f n 1 ··· n k , b ) Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) A uniform version of Hypothesis H ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Conjecture for friable values of polynomials � �� x / n 1 · · · n k 1 � Statement of the conjecture × 1 + O . Reduction to convenient log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) log x polynomials Translation into prime values of polynomials Shepherding the local factors Sums of multiplicative functions � Summary H ( f n 1 ··· n k , b ) Next: concentrate on b ∈R ( f ; n 1 ,..., n k )

  76. Friable values of Nice sums over local solutions polynomials Greg Martin Recall Introduction � − k � Friable integers � 1 − 1 1 − σ ( f ; p ) � � Friable values of polynomials H ( f ) = p p Bounds for friable p values of polynomials How friable can values of special polynomials be? How friable can values of general Recall polynomials be? Can we have lots of friable values? Conjecture for prime σ ( f ; p ) = { a (mod p ) : f ( a ) ≡ 0 (mod p ) } values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Proposition values of polynomials Statement of the conjecture � Reduction to convenient H ( f n 1 ··· n k , b ) = H ( f ) g 1 ( n 1 ) · · · g k ( n k ) , where polynomials Translation into prime values of b ∈R ( f ; n 1 ,..., n k ) polynomials Shepherding the local factors Sums of multiplicative functions � − 1 � σ ( f j ; p ν ) − σ ( f j ; p ν + 1 ) � 1 − σ ( f ; p ) � Summary � g j ( n j ) = . p p p ν � n j

  77. Friable values of Nice sums over local solutions polynomials Greg Martin Recall Introduction � − k � Friable integers � 1 − 1 1 − σ ( f ; p ) � � Friable values of polynomials H ( f ) = p p Bounds for friable p values of polynomials How friable can values of special polynomials be? How friable can values of general Recall polynomials be? Can we have lots of friable values? Conjecture for prime σ ( f ; p ) = { a (mod p ) : f ( a ) ≡ 0 (mod p ) } values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Proposition values of polynomials Statement of the conjecture � Reduction to convenient H ( f n 1 ··· n k , b ) = H ( f ) g 1 ( n 1 ) · · · g k ( n k ) , where polynomials Translation into prime values of b ∈R ( f ; n 1 ,..., n k ) polynomials Shepherding the local factors Sums of multiplicative functions � − 1 � σ ( f j ; p ν ) − σ ( f j ; p ν + 1 ) � 1 − σ ( f ; p ) � Summary � g j ( n j ) = . p p p ν � n j

  78. Friable values of Nice sums over local solutions polynomials Greg Martin Recall Introduction � − k � Friable integers � 1 − 1 1 − σ ( f ; p ) � � Friable values of polynomials H ( f ) = p p Bounds for friable p values of polynomials How friable can values of special polynomials be? How friable can values of general Recall polynomials be? Can we have lots of friable values? � R ( f ; n 1 , . . . , n k ) = b (mod n 1 · · · n k ) : Conjecture for prime values of polynomials � n 1 | f 1 ( b ) , n 2 | f 2 ( b ) , . . . , n k | f k ( b ) Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Proposition values of polynomials Statement of the conjecture � Reduction to convenient H ( f n 1 ··· n k , b ) = H ( f ) g 1 ( n 1 ) · · · g k ( n k ) , where polynomials Translation into prime values of b ∈R ( f ; n 1 ,..., n k ) polynomials Shepherding the local factors Sums of multiplicative functions � − 1 � σ ( f j ; p ν ) − σ ( f j ; p ν + 1 ) � 1 − σ ( f ; p ) � Summary � g j ( n j ) = . p p p ν � n j

  79. Friable values of Nice sums over local solutions polynomials Greg Martin Recall Introduction � − k � Friable integers � 1 − 1 1 − σ ( f ; p ) � � Friable values of polynomials H ( f ) = p p Bounds for friable p values of polynomials How friable can values of special polynomials be? How friable can values of general Recall polynomials be? Can we have lots of friable values? � R ( f ; n 1 , . . . , n k ) = b (mod n 1 · · · n k ) : Conjecture for prime values of polynomials � n 1 | f 1 ( b ) , n 2 | f 2 ( b ) , . . . , n k | f k ( b ) Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Proposition values of polynomials Statement of the conjecture � Reduction to convenient H ( f n 1 ··· n k , b ) = H ( f ) g 1 ( n 1 ) · · · g k ( n k ) , where polynomials Translation into prime values of b ∈R ( f ; n 1 ,..., n k ) polynomials Shepherding the local factors Sums of multiplicative functions � − 1 � σ ( f j ; p ν ) − σ ( f j ; p ν + 1 ) � 1 − σ ( f ; p ) � Summary � g j ( n j ) = . p p p ν � n j

  80. Friable values of Nice sums over local solutions polynomials Greg Martin Recall Introduction � − k � Friable integers � 1 − 1 1 − σ ( f ; p ) � � Friable values of polynomials H ( f ) = p p Bounds for friable p values of polynomials How friable can values of special polynomials be? How friable can values of general Proving this proposition . . . polynomials be? Can we have lots of friable values? . . . is fun, actually, involving the Chinese remainder theorem, Conjecture for prime values of polynomials counting lifts of local solutions (Hensel’s lemma), and so on. Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Conjecture for friable Proposition values of polynomials Statement of the conjecture � Reduction to convenient H ( f n 1 ··· n k , b ) = H ( f ) g 1 ( n 1 ) · · · g k ( n k ) , where polynomials Translation into prime values of b ∈R ( f ; n 1 ,..., n k ) polynomials Shepherding the local factors Sums of multiplicative functions � − 1 � σ ( f j ; p ν ) − σ ( f j ; p ν + 1 ) � 1 − σ ( f ; p ) � Summary � g j ( n j ) = . p p p ν � n j

  81. Friable values of polynomials For f = f 1 . . . f k and x and y sufficiently large, Greg Martin � � � � � M ( f ; x , y ) = · · · H ( f n 1 ··· n k , b ) Introduction Friable integers n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) Friable values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Bounds for friable x / n 1 · · · n k � 1 �� values of polynomials � × 1 + O How friable can values of special log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) log x polynomials be? How friable can values of general polynomials be? � 1 �� � Can we have lots of friable values? = xH ( f ) 1 + O log x Conjecture for prime values of polynomials g 1 ( n 1 ) · · · g k ( n k ) / n 1 · · · n k Schinzel’s “Hypothesis H” � � × · · · log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) . (Bateman–Horn conjecture) A uniform version of Hypothesis H n 1 ≤ ξ 1 / y n k ≤ ξ k / y Conjecture for friable ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) values of polynomials Statement of the conjecture Reduction to convenient polynomials g 1 ( n 1 ) · · · g k ( n k ) � � Therefore: consider · · · Translation into prime values of polynomials n 1 · · · n k Shepherding the local factors n 1 ≤ ξ 1 / y n k ≤ ξ k / y Sums of multiplicative functions ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Summary (take care of logarithms later, via partial summation)

  82. Friable values of polynomials For f = f 1 . . . f k and x and y sufficiently large, Greg Martin � � � � � M ( f ; x , y ) = · · · H ( f n 1 ··· n k , b ) Introduction Friable integers n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) Friable values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Bounds for friable x / n 1 · · · n k � 1 �� values of polynomials � × 1 + O How friable can values of special log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) log x polynomials be? How friable can values of general polynomials be? � 1 �� � Can we have lots of friable values? = xH ( f ) 1 + O log x Conjecture for prime values of polynomials g 1 ( n 1 ) · · · g k ( n k ) / n 1 · · · n k Schinzel’s “Hypothesis H” � � × · · · log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) . (Bateman–Horn conjecture) A uniform version of Hypothesis H n 1 ≤ ξ 1 / y n k ≤ ξ k / y Conjecture for friable ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) values of polynomials Statement of the conjecture Reduction to convenient polynomials g 1 ( n 1 ) · · · g k ( n k ) � � Therefore: consider · · · Translation into prime values of polynomials n 1 · · · n k Shepherding the local factors n 1 ≤ ξ 1 / y n k ≤ ξ k / y Sums of multiplicative functions ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Summary (take care of logarithms later, via partial summation)

  83. Friable values of polynomials For f = f 1 . . . f k and x and y sufficiently large, Greg Martin � � � � � M ( f ; x , y ) = · · · H ( f n 1 ··· n k , b ) Introduction Friable integers n 1 ≤ ξ 1 / y n k ≤ ξ k / y b ∈R ( f ; n 1 ,..., n k ) Friable values of polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Bounds for friable x / n 1 · · · n k � 1 �� values of polynomials � × 1 + O How friable can values of special log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) log x polynomials be? How friable can values of general polynomials be? � 1 �� � Can we have lots of friable values? = xH ( f ) 1 + O log x Conjecture for prime values of polynomials g 1 ( n 1 ) · · · g k ( n k ) / n 1 · · · n k Schinzel’s “Hypothesis H” � � × · · · log ( ξ 1 / n 1 ) · · · log ( ξ k / n k ) . (Bateman–Horn conjecture) A uniform version of Hypothesis H n 1 ≤ ξ 1 / y n k ≤ ξ k / y Conjecture for friable ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) values of polynomials Statement of the conjecture Reduction to convenient polynomials g 1 ( n 1 ) · · · g k ( n k ) � � Therefore: consider · · · Translation into prime values of polynomials n 1 · · · n k Shepherding the local factors n 1 ≤ ξ 1 / y n k ≤ ξ k / y Sums of multiplicative functions ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Summary First: consider more general sums of multiplicative functions

  84. Friable values of Multiplicative functions: one-variable sums polynomials Greg Martin Definition Let’s say a multiplicative function g ( n ) is α on average if it Introduction Friable integers takes nonnegative values and Friable values of polynomials Bounds for friable g ( p ) log p values of polynomials � ∼ α log w . How friable can values of special p polynomials be? p ≤ w How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Note: we really need upper bounds on g ( p ν ) as well . . . values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Lemma Conjecture for friable values of polynomials If the multiplicative function g ( n ) is α on average, then Statement of the conjecture Reduction to convenient polynomials g ( n ) � Translation into prime values of ∼ c ( g )( log t ) α , polynomials n Shepherding the local factors n ≤ t Sums of multiplicative functions Summary � α � + g ( p 2 ) � 1 − 1 1 + g ( p ) � � where c ( g ) = + · · · . p 2 p p p

  85. Friable values of Multiplicative functions: one-variable sums polynomials Greg Martin Definition Let’s say a multiplicative function g ( n ) is α on average if it Introduction Friable integers takes nonnegative values and Friable values of polynomials Bounds for friable g ( p ) log p values of polynomials � ∼ α log w . How friable can values of special p polynomials be? p ≤ w How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Note: we really need upper bounds on g ( p ν ) as well . . . values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Lemma Conjecture for friable values of polynomials If the multiplicative function g ( n ) is α on average, then Statement of the conjecture Reduction to convenient polynomials g ( n ) � Translation into prime values of ∼ c ( g )( log t ) α , polynomials n Shepherding the local factors n ≤ t Sums of multiplicative functions Summary � α � + g ( p 2 ) � 1 − 1 1 + g ( p ) � � where c ( g ) = + · · · . p 2 p p p

  86. Friable values of Multiplicative functions: one-variable sums polynomials Greg Martin Definition Let’s say a multiplicative function g ( n ) is α on average if it Introduction Friable integers takes nonnegative values and Friable values of polynomials Bounds for friable g ( p ) log p values of polynomials � ∼ α log w . How friable can values of special p polynomials be? p ≤ w How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Note: we really need upper bounds on g ( p ν ) as well . . . values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Lemma Conjecture for friable values of polynomials If the multiplicative function g ( n ) is α on average, then Statement of the conjecture Reduction to convenient polynomials g ( n ) � Translation into prime values of ∼ c ( g )( log t ) α , polynomials n Shepherding the local factors n ≤ t Sums of multiplicative functions Summary � α � + g ( p 2 ) � 1 − 1 1 + g ( p ) � � where c ( g ) = + · · · . p 2 p p p

  87. Friable values of Multiplicative functions: one-variable sums polynomials Greg Martin Definition Let’s say a multiplicative function g ( n ) is α on average if it Introduction Friable integers takes nonnegative values and Friable values of polynomials Bounds for friable g ( p ) log p values of polynomials � ∼ α log w . How friable can values of special p polynomials be? p ≤ w How friable can values of general polynomials be? Can we have lots of friable values? Conjecture for prime Note: we really need upper bounds on g ( p ν ) as well . . . values of polynomials Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H Lemma Conjecture for friable values of polynomials If the multiplicative function g ( n ) is α on average, then Statement of the conjecture Reduction to convenient polynomials g ( n ) � Translation into prime values of ∼ c ( g )( log t ) α , polynomials n Shepherding the local factors n ≤ t Sums of multiplicative functions Summary � α � + g ( p 2 ) � 1 − 1 1 + g ( p ) � � where c ( g ) = + · · · . p 2 p p p

  88. Friable values of Multiplicative functions: more variables polynomials Greg Martin From previous slide � α � Introduction + g ( p 2 ) � 1 − 1 1 + g ( p ) � � c ( g ) = + · · · Friable integers p 2 p p Friable values of polynomials p Bounds for friable values of polynomials By the lemma on the previous slide, we easily get: How friable can values of special polynomials be? How friable can values of general polynomials be? Proposition Can we have lots of friable values? Conjecture for prime If the multiplicative functions g 1 ( n ) , . . . , g k ( n ) are each 1 on values of polynomials average, then Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H g 1 ( n 1 ) · · · g k ( n k ) Conjecture for friable � � c ( g 1 ) · · · c ( g k )( log t ) k . · · · ∼ values of polynomials n 1 · · · n k Statement of the conjecture n 1 ≤ t n k ≤ t Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors However, we need the analogous sum with the coprimality Sums of multiplicative functions condition ( n i , n j ) = 1. (This is where K > 1 makes life Summary harder!)

  89. Friable values of Multiplicative functions: more variables polynomials Greg Martin From previous slide � α � Introduction + g ( p 2 ) � 1 − 1 1 + g ( p ) � � c ( g ) = + · · · Friable integers p 2 p p Friable values of polynomials p Bounds for friable values of polynomials By the lemma on the previous slide, we easily get: How friable can values of special polynomials be? How friable can values of general polynomials be? Proposition Can we have lots of friable values? Conjecture for prime If the multiplicative functions g 1 ( n ) , . . . , g k ( n ) are each 1 on values of polynomials average, then Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H g 1 ( n 1 ) · · · g k ( n k ) Conjecture for friable � � c ( g 1 ) · · · c ( g k )( log t ) k . · · · ∼ values of polynomials n 1 · · · n k Statement of the conjecture n 1 ≤ t n k ≤ t Reduction to convenient polynomials Translation into prime values of polynomials Shepherding the local factors However, we need the analogous sum with the coprimality Sums of multiplicative functions condition ( n i , n j ) = 1. (This is where K > 1 makes life Summary harder!)

  90. Friable values of Multiplicative functions: more variables polynomials Greg Martin From previous slide � α � Introduction + g ( p 2 ) � 1 − 1 1 + g ( p ) � � c ( g ) = + · · · Friable integers p 2 p p Friable values of polynomials p Bounds for friable values of polynomials We get: How friable can values of special polynomials be? How friable can values of general polynomials be? Proposition Can we have lots of friable values? Conjecture for prime If the multiplicative functions g 1 ( n ) , . . . , g k ( n ) are each 1 on values of polynomials average, then Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H g 1 ( n 1 ) · · · g k ( n k ) Conjecture for friable � � ∼ c ( g 1 + · · · + g k )( log t ) k . · · · values of polynomials n 1 · · · n k Statement of the conjecture n 1 ≤ t n k ≤ t Reduction to convenient polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Translation into prime values of polynomials Shepherding the local factors However, we need the analogous sum with the coprimality Sums of multiplicative functions condition ( n i , n j ) = 1. (This is where K > 1 makes life Summary harder!)

  91. Friable values of Multiplicative functions: more variables polynomials Greg Martin From previous slide � α � Introduction + g ( p 2 ) � 1 − 1 1 + g ( p ) � � c ( g ) = + · · · Friable integers p 2 p p Friable values of polynomials p Bounds for friable values of polynomials We get: How friable can values of special polynomials be? How friable can values of general polynomials be? Proposition Can we have lots of friable values? Conjecture for prime If the multiplicative functions g 1 ( n ) , . . . , g k ( n ) are each 1 on values of polynomials average, then Schinzel’s “Hypothesis H” (Bateman–Horn conjecture) A uniform version of Hypothesis H g 1 ( n 1 ) · · · g k ( n k ) Conjecture for friable � � ∼ c ( g 1 + · · · + g k )( log t ) k . · · · values of polynomials n 1 · · · n k Statement of the conjecture n 1 ≤ t n k ≤ t Reduction to convenient polynomials ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Translation into prime values of polynomials Shepherding the local factors However, we need the analogous sum with the coprimality Sums of multiplicative functions condition ( n i , n j ) = 1. (This is where K > 1 makes life Summary harder!) Never mind that g 1 + · · · + g k isn’t multiplicative!

  92. Friable values of Partial summation: return of the logs polynomials Greg Martin The proposition on the previous slide: Introduction . . . gives, after a k -fold partial summation argument: Friable integers Friable values of polynomials Proposition Bounds for friable values of polynomials If the multiplicative functions g 1 ( n ) , . . . , g k ( n ) are each 1 on How friable can values of special polynomials be? average, then How friable can values of general polynomials be? Can we have lots of friable values? g 1 ( n 1 ) · · · g k ( n k ) Conjecture for prime � � · · · values of polynomials n 1 · · · n k Schinzel’s “Hypothesis H” n 1 ≤ ξ 1 / y n k ≤ ξ k / y (Bateman–Horn conjecture) A uniform version of Hypothesis H ( n i , n j )= 1 ( 1 ≤ i < j ≤ k ) Conjecture for friable k values of polynomials log ξ j � ∼ c ( g 1 + · · · + g k ) y . Statement of the conjecture Reduction to convenient polynomials j = 1 Translation into prime values of polynomials � − 1 � σ ( f j ; p ) − σ ( f j ; p 2 ) Shepherding the local factors 1 − σ ( f ; p ) � � For our functions, g j ( p ) = Sums of multiplicative functions p p � 1 � �� Summary = σ ( f j ; p ) 1 + O , and σ ( f j ; p ) is indeed 1 on average by p the prime ideal theorem.

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