Primes in arithmetic progressions to large moduli James Maynard - PowerPoint PPT Presentation

Primes in arithmetic progressions to large moduli James Maynard University of Oxford Second Symposium in Analytic Number Theory, Cetraro July 2019 James Maynard Primes in arithmetic progressions to large moduli

Introduction How many primes are less than x and congruent to a ( mod q ) ? James Maynard Primes in arithmetic progressions to large moduli

Introduction How many primes are less than x and congruent to a ( mod q ) ? Theorem (Siegel-Walfisz) If q ≤ (log x ) A and gcd( a , q ) = 1 then π ( x ; q , a ) = ( 1 + o ( 1 )) π ( x ) φ ( q ) . Theorem (GRH Bound) Assume GRH. If q ≤ x 1 / 2 − ǫ and gcd( a , q ) = 1 then π ( x ; q , a ) = ( 1 + o ( 1 )) π ( x ) φ ( q ) . Conjecture (Montgomery) If q ≤ x 1 − ǫ and gcd( a , q ) = 1 then π ( x ; q , a ) = ( 1 + o ( 1 )) π ( x ) φ ( q ) . James Maynard Primes in arithmetic progressions to large moduli

Introduction II Often we don’t need such a statement to be true for every q , just for most q . James Maynard Primes in arithmetic progressions to large moduli

Introduction II Often we don’t need such a statement to be true for every q , just for most q . Theorem (Bombieri-Vinogradov) Let Q < x 1 / 2 − ǫ . Then for any A � π ( x ; q , a ) − π ( x ) x � � � sup � � ≪ A � � � φ ( q ) (log x ) A ( a , q )= 1 q ∼ Q Corollary For most q ≤ x 1 / 2 − ǫ , we have π ( x ; q , a ) = ( 1 + o ( 1 )) π ( x ) φ ( q ) for every a with gcd( a , q ) = 1 . James Maynard Primes in arithmetic progressions to large moduli

Introduction II Often we don’t need such a statement to be true for every q , just for most q . Theorem (Bombieri-Vinogradov) Let Q < x 1 / 2 − ǫ . Then for any A � π ( x ; q , a ) − π ( x ) x � � � sup � � � ≪ A � � φ ( q ) (log x ) A ( a , q )= 1 q ∼ Q Corollary For most q ≤ x 1 / 2 − ǫ , we have π ( x ; q , a ) = ( 1 + o ( 1 )) π ( x ) φ ( q ) for every a with gcd( a , q ) = 1 . From the point of view of e.g. sieve methods, this is essentially as good as the Riemann Hypothesis! James Maynard Primes in arithmetic progressions to large moduli

Beyond GRH Pioneering work by Bombieri, Fouvry, Friedlander, Iwaniec went beyond the x 1 / 2 barrier in special circumstances. James Maynard Primes in arithmetic progressions to large moduli

Beyond GRH Pioneering work by Bombieri, Fouvry, Friedlander, Iwaniec went beyond the x 1 / 2 barrier in special circumstances. Theorem (BFI1) Fix a. Then we have (uniformly in θ ) � ≪ a ( θ − 1 / 2 ) 2 x (log log x ) O ( 1 ) � π ( x ; q , a ) − π ( x ) x � � � � � + . � � log 3 x φ ( q ) log x q ∼ x θ ( q , a )= 1 This is non-trivial when θ is very close to 1 / 2. Theorem (BFI2) Fix a. Let λ ( q ) be ‘well-factorable’. Then we have π ( x ; q , a ) − π ( x ) x � � � λ ( q ) ≪ a , A . log A x φ ( q ) q ∼ x 4 / 7 − ǫ ( q , a )= 1 This is often an adequate substitute for BV with exponent 4 / 7! James Maynard Primes in arithmetic progressions to large moduli

Beyond GRH II More recently, Zhang went beyond x 1 / 2 for smooth/friable moduli. Theorem (Zhang,Polymath) � π ( x ; q , a ) − π ( x ) x � � � � � ≪ A � � � (log x ) A φ ( q ) q ≤ x 1 / 2 + 7 / 300 − ǫ p | q ⇒ p ≤ x ǫ 2 ( q , a )= 1 The implied constant is independent of a . James Maynard Primes in arithmetic progressions to large moduli

New results Theorem (M.) Let δ < 1 / 42 and Q δ := { q ∼ x 1 / 2 + δ : ∃ d | q s.t. x 2 δ + ǫ < d < x 1 / 14 − δ } . x (log log x ) O ( 1 ) � π ( x ; q , a ) − π ( x ) � � � � � � ≪ A . � � log 5 x φ ( q ) q ∈Q δ ( q , a )= 1 James Maynard Primes in arithmetic progressions to large moduli

New results Theorem (M.) Let δ < 1 / 42 and Q δ := { q ∼ x 1 / 2 + δ : ∃ d | q s.t. x 2 δ + ǫ < d < x 1 / 14 − δ } . x (log log x ) O ( 1 ) � π ( x ; q , a ) − π ( x ) � � � � � � ≪ A . � � log 5 x φ ( q ) q ∈Q δ ( q , a )= 1 Corollary Let δ < 1 / 42 . For ( 100 − O ( δ ))% of q ∼ x 1 / 2 + δ we have π ( x ; q , a ) = ( 1 + o ( 1 )) π ( x ) φ ( q ) Corollary x (log log x ) O ( 1 ) π ( x ) � � � � � � π ( x ; q 1 q 2 , a ) − � � ≪ a � � log 5 x φ ( q 1 q 2 ) q 1 ∼ x 1 / 21 q 2 ∼ x 10 / 21 − ǫ ( q 1 q 2 , a )= 1 James Maynard Primes in arithmetic progressions to large moduli

New Results II Theorem (M.) Let λ ( q ) be ‘very well factorable’. Then we have π ( x ; q , a ) − π ( x ) x � � � λ ( q ) ≪ a , A (log x ) A . φ ( q ) q ≤ x 3 / 5 − ǫ ( q , a )= 1 The β -sieve weights are ‘very well factorable’ for β ≥ 2. James Maynard Primes in arithmetic progressions to large moduli

New Results II Theorem (M.) Let λ ( q ) be ‘very well factorable’. Then we have π ( x ; q , a ) − π ( x ) x � � � λ ( q ) ≪ a , A (log x ) A . φ ( q ) q ≤ x 3 / 5 − ǫ ( q , a )= 1 The β -sieve weights are ‘very well factorable’ for β ≥ 2. Corollary Let λ + ( d ) be sieve weights for the linear sieve. Then π ( x ; q , a ) − π ( x ) x � � � λ + ( q ) ≪ (log x ) A . φ ( q ) q ≤ x 7 / 12 − ǫ ( q , a )= 1 James Maynard Primes in arithmetic progressions to large moduli

Comparison Result Size of q Type of q Proportion of q x 1 / 2 + o ( 1 ) BFI1 All ( 100 − δ )% x 4 / 7 − ǫ δ % BFI2 Factorable x 1 / 2 + 7 / 300 − ǫ Zhang Factorable δ % x 11 / 21 − ǫ ( 100 − δ )% M1 Partially Factorable x 3 / 5 − ǫ M2 Factorable δ % Result Coefficients Residue class Cancellation o ( 1 ) BFI1 Absolute values Fixed log A x BFI2 Factorable weights Fixed log A x Zhang Absolute values Uniform log 5 − ǫ x M1 Absolute values Fixed log A x M2 Factorable weights Fixed Note that 3 / 5 > 4 / 7 > 11 / 21 > 1 / 2 + 7 / 300. James Maynard Primes in arithmetic progressions to large moduli

Proof overview The overall proof follows the same lines as previous approaches: Apply a combinatorial decomposition to Λ( n ) 1 James Maynard Primes in arithmetic progressions to large moduli

Proof overview The overall proof follows the same lines as previous approaches: Apply a combinatorial decomposition to Λ( n ) 1 Reduce the problem to estimating exponenital sums of 2 convolutions James Maynard Primes in arithmetic progressions to large moduli

Proof overview The overall proof follows the same lines as previous approaches: Apply a combinatorial decomposition to Λ( n ) 1 Reduce the problem to estimating exponenital sums of 2 convolutions Apply different techniques in different ranges to estimate 3 exponential sums Bounds from the spectral theory of automorphic forms (Kuznetsov Trace Formula) Bounds from Algebraic Geometry (Weil bound/Deligne bounds) James Maynard Primes in arithmetic progressions to large moduli

Proof overview The overall proof follows the same lines as previous approaches: Apply a combinatorial decomposition to Λ( n ) 1 Reduce the problem to estimating exponenital sums of 2 convolutions Apply different techniques in different ranges to estimate 3 exponential sums Bounds from the spectral theory of automorphic forms (Kuznetsov Trace Formula) Bounds from Algebraic Geometry (Weil bound/Deligne bounds) Ensure that (essentially) all ranges are covered. 4 James Maynard Primes in arithmetic progressions to large moduli

Proof overview The overall proof follows the same lines as previous approaches: Apply a combinatorial decomposition to Λ( n ) 1 Reduce the problem to estimating exponenital sums of 2 convolutions Apply different techniques in different ranges to estimate 3 exponential sums Bounds from the spectral theory of automorphic forms (Kuznetsov Trace Formula) Bounds from Algebraic Geometry (Weil bound/Deligne bounds) Ensure that (essentially) all ranges are covered. 4 *Combine Zhang-style estimates with Kloostermania* James Maynard Primes in arithmetic progressions to large moduli

Bad products Let us recall the situation when q ∼ x 1 / 2 + δ where δ > 0 is fixed but small. Using BFI proof ideas: Heath-Brown Identity/Sieve methods reduces to considering 1 products of few prime factors James Maynard Primes in arithmetic progressions to large moduli

Bad products Let us recall the situation when q ∼ x 1 / 2 + δ where δ > 0 is fixed but small. Using BFI proof ideas: Heath-Brown Identity/Sieve methods reduces to considering 1 products of few prime factors Working through the BFI argument their proof can essentially 2 handle all such numbers except for Products p 1 p 2 p 3 p 4 p 5 of 5 primes with p i = x 1 / 5 + O ( δ ) Products p 1 p 2 p 3 p 4 of 4 primes with p i = x 1 / 4 + O ( δ ) James Maynard Primes in arithmetic progressions to large moduli

Bad products Let us recall the situation when q ∼ x 1 / 2 + δ where δ > 0 is fixed but small. Using BFI proof ideas: Heath-Brown Identity/Sieve methods reduces to considering 1 products of few prime factors Working through the BFI argument their proof can essentially 2 handle all such numbers except for Products p 1 p 2 p 3 p 4 p 5 of 5 primes with p i = x 1 / 5 + O ( δ ) Products p 1 p 2 p 3 p 4 of 4 primes with p i = x 1 / 4 + O ( δ ) BFI result follows on noting that these terms are only a O ( δ ) proportion of the terms. We can concentrate on these ‘bad products’ . James Maynard Primes in arithmetic progressions to large moduli