Marseille 2019 On the digits of primes In memoriam Christian - PowerPoint PPT Presentation

Marseille 2019 On the digits of primes In memoriam Christian MAUDUIT Jo ¨ e l RIVAT Institut de Math ´ e matiques de Marseille, Universit ´ e d’Aix-Marseille. 1

2

Digits Let q � 2 be an integer. Any n ∈ N can be written � ε j ( n ) q j , n = ε j ( n ) ∈ { 0 , . . . , q − 1 } . j � 0 The sum of digits function � s( n ) = ε j ( n ) j � 0 has been studied in many directions: ergodicity, finite automata, dynamical systems, harmonic analysis, number theory, etc. Mahler, 1927: For q = 2 , the sequence   �  1 ( − 1) s ( n ) ( − 1) s ( n + k )  N n<N N � 1 converges for all k ∈ N and its limit is different from zero for infinitely many k ’s. 3

Histogram of the sum of binary digits of integers (binomial distribution) card { n � 10 10 , s( n ) = k } · 10 9 1 . 4 1 . 2 1 0 . 8 0 . 6 0 . 4 0 . 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 k 4

Gelfond’s paper Gelfond, 1968: The sum of digits in base q � 2 is well distributed along arithmetic progressions. More precisely given m � 2 with ( m, q − 1) = 1 , there exists an explicit σ m > 0 such that ∀ m ′ ∈ N ∗ , ∀ ( n ′ , s ) ∈ Z 2 , � x mm ′ + O ( x 1 − σ m ) . 1 = n � x n ≡ n ′ mod m ′ s( n ) ≡ s mod m A.O. Gelfond 5

Gelfond’s problems, 1968 1. Evaluate the number of prime numbers p � x such that s( p ) ≡ a mod m . 2. Evaluate the number of integers n � x such that s( P ( n )) ≡ a mod m , where P is a suitable polynomial [for example P ( n ) = n 2 ] . 6

Histogram of the sum of binary digits of prime numbers card { p � 10 10 , s( p ) = k } · 10 7 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 k 7

Sum of binary digits of prime numbers in residue classes modulo 2 modulo 3 modulo 4 2 . 5 · 10 8 · 10 8 · 10 8 1 . 2 1 . 5 2 1 0 . 8 1 1 . 5 0 . 6 1 0 . 4 0 . 5 0 . 5 0 . 2 0 0 0 0 1 0 1 2 0 1 2 3 modulo 5 modulo 6 modulo 7 1 · 10 8 · 10 7 · 10 7 8 0 . 8 6 6 0 . 6 4 4 0 . 4 2 2 0 . 2 0 0 0 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 8

Partial results Fouvry–Mauduit (1996): � � 1 � C ( q, m ) 1 . log log x n � x n � x n = p or n = p 1 p 2 n = p or n = p 1 p 2 s( n ) ≡ a mod m Dartyge–Tenenbaum (2005): For r � 2 , � � C ( q, m, r ) 1 � 1 . log log x log log log x n � x n � x n = p 1 ...p r n = p 1 ...p r s( n ) ≡ a mod m 9

Gelfond’s conjecture for primes Mauduit-Rivat, 2010: If ( q − 1) α ∈ R \ Z , there exists C q ( α ) > 0 and σ q ( α ) > 0 , � � � � � � � � � C q ( α ) x 1 − σ q ( α ) . � � exp(2 iπα s( p )) � � � p � x Hence • For q � 2 the sequence ( α s( p n )) n � 1 is equidistributed modulo 1 if and only if α ∈ R \ Q (here ( p n ) n � 1 denotes the sequence of prime numbers). • (Gelfond’s problem): for q � 2 , m � 2 such that ( m, q − 1) = 1 and a ∈ Z , � � 1 1 ∼ 1 ( x → + ∞ ) . m p � x p � x s( p ) ≡ a mod m 10

Histogram of local result for prime numbers card { p � 10 10 , s( p ) = k } · 10 7 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 20 21 22 23 24 25 26 27 28 29 30 31 32 33 k 11

Local result for prime numbers Drmota-Mauduit-Rivat, 2009: uniformly for all integers k � 0 with ( k, q − 1) = 1 # { p � x : s( p ) = k } � � − ( k − µ q log q x ) 2 � � q − 1 π ( x ) + O ((log x ) − 1 2 + ε ) = � exp , 2 σ 2 ϕ ( q − 1) 2 πσ 2 q log q x q log q x where q = q 2 − 1 µ q = q − 1 σ 2 , 2 12 and ε > 0 is arbitrary but fixed. Such a local result was considered by Erd˝ os as “ hopelessly difficult ” . 12

Discrete Fourier Transform For f : N → C and λ ∈ N we define a 2 λ -periodic function f λ : Z → C by ∀ u ∈ { 0 , . . . , 2 λ − 1 } , f λ ( u ) = f ( u ) and its Discrete Fourier Transform � − 2 iπut � � f λ ( t ) = 1 � f λ ( u ) exp . 2 λ 2 λ 0 � u< 2 λ   1 / 2 � � � � � 2 � � � � �� ��   � 2 = f λ ( h ) = 1 . By orthogonality f λ � 0 � h< 2 λ � � � � � � � � �� �� A non-trivial upper bound for f λ � ∞ or f λ � 1 is a challenging problem. � � � � �� � 1 = O (2 ηλ ) with η < 1 / 2 was crucial for solving Gelfond’s conjecture. Getting f λ 13

The Rudin-Shapiro sequence � j � 1 ε j − 1 ( n ) ε j ( n ) . Let f ( n ) = ( − 1) � � − 2 iπut/ 2 λ �� f λ ( t ) = 2 − λ · Schapiro polynomial For λ ∈ N , we have � exp , hence � � 1 − λ � � �� 2 . f λ � ∞ � 2 � � � � �� � 2 = 1 , by Cauchy-Schwarz it is easy to deduce that Since f λ � � λ − 1 λ � � �� 2 f λ � 1 � 2 2 � 2 � � � � � 1 = O (2 ηλ ) with η < 1 �� The proof for the sum of digits function requires f λ 2 . This is not satisfied for the Rudin-Shapiro sequence !!! 14

Histogram of “ Rudin-Shapiro sums ” of prime numbers � � p � 10 10 , � card j � 1 ε j − 1 ( p ) ε j ( p ) = k · 10 7 6 5 . 5 5 4 . 5 4 3 . 5 3 2 . 5 2 1 . 5 1 0 . 5 0 − 0 . 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 k 15

“ Rudin-Shapiro sums ” of prime numbers in residue classes modulo 2 modulo 3 modulo 4 2 . 5 · 10 8 · 10 8 · 10 8 1 . 2 1 . 5 2 1 0 . 8 1 . 5 1 0 . 6 1 0 . 4 0 . 5 0 . 5 0 . 2 0 0 0 0 1 0 1 2 0 1 2 3 modulo 5 modulo 6 modulo 7 1 · 10 8 · 10 7 · 10 7 8 6 0 . 8 6 0 . 6 4 4 0 . 4 2 2 0 . 2 0 0 0 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 6 16

Rudin-Shapiro sequences of order δ Let δ ∈ N and β δ ( n ) the number of occurencies of patterns 1 ∗ · · · ∗ 1 , i.e. of the form 1 w 1 � �� � δ (where w ∈ { 0 , 1 } δ ) in the representation of n : � β δ ( n ) = ε k − δ − 1 ( n ) ε k ( n ) . k � δ +1 Mauduit-Rivat, 2015: for any δ ∈ N , α ∈ R , ϑ ∈ R and x � 2 , there exists explicit constants C ( δ ) and σ ( α ) > 0 such that � � � � � � � 11 4 x 1 − σ ( α ) � � Λ( n ) e ( β δ ( n ) α + ϑn ) � � C ( δ ) (log x ) � � � n � x and � � � � � � � 11 4 x 1 − σ ( α ) . � � µ ( n ) e ( β δ ( n ) α + ϑn ) � � C ( δ ) (log x ) � � � n � x 17

Rudin-Shapiro sequences of degree d Let d ∈ N with d � 2 and b d ( n ) denote the number of occurencies of 1 · · · 1 i.e. blocks of d � �� � d consecutive 1 ’s in the representation of n in base 2 : � b d ( n ) = ε k − d +1 ( n ) · · · ε k ( n ) . k � d − 1 Mauduit-Rivat, 2015: for any d ∈ N with d � 2 , α ∈ R , ϑ ∈ R and x � 2 there exist an explicit constant σ ( d, α ) > 0 such that � � � � � � � 11 4 x 1 − σ ( d,α ) , � � � ≪ (log x ) Λ( n ) e ( b d ( n ) α + ϑn ) � � � n � x � � � � � � � 11 4 x 1 − σ ( d,α ) . � � µ ( n ) e ( b d ( n ) α + ϑn ) � ≪ (log x ) � � � n � x 18

General result – Definitions Let U = { z ∈ C , | z | = 1 } . Definition 1 A function f : N → U has the carry property if, uniformly for ( λ, κ, ρ ) ∈ N 3 with ρ < λ , the number of integers 0 � ℓ < q λ such that there exists ( k 1 , k 2 ) ∈ { 0 , . . . , q κ − 1 } 2 with f ( ℓq κ + k 1 + k 2 ) f ( ℓq κ + k 1 ) � = f κ + ρ ( ℓq κ + k 1 + k 2 ) f κ + ρ ( ℓq κ + k 1 ) is at most O ( q λ − ρ ) where the implied constant may depend only on q and f . Definition 2 Given a non decreasing function γ : R → R satisfying lim λ → + ∞ γ ( λ ) = + ∞ and c > 0 we denote by F γ,c the set of functions f : N → U such that for ( κ, λ ) ∈ N 2 with κ � cλ and t ∈ R : � � � � � � � � � q − λ f ( uq κ ) e ( − ut ) � q − γ ( λ ) . � � � � � � 0 � u<q λ 19

General result Let γ : R → R be a non decreasing function satisfying lim λ → + ∞ γ ( λ ) = + ∞ , c � 10 and f : N → U be a function satisfying Definition 1 and f ∈ F γ,c in Definition 2. Then for any θ ∈ R we have � � � � � � � � ≪ c 1 ( q )(log x ) c 2 ( q ) x q − γ (2 ⌊ (log x ) / 80 log q ⌋ ) / 20 , � � Λ( n ) f ( n ) e ( θn ) � � � n � x with explicit c 1 ( q ) and c 2 ( q ) . Of course the same estimate holds if we replace the von Mangoldt function Λ by the M ¨ o bius function µ . M¨ ullner, 2018, has extended this result to all automatic sequences ! 20

Primes in two bases Drmota-Mauduit-Rivat, 2019+: If f is a strongly q 1 -multiplicative function and g a strongly q 2 -multiplicative function such that ( q 1 , q 2 ) = 1 and f is is not of the form n �→ e( kn/ ( q 1 − 1)) with k ∈ Z , then we have uniformly for ϑ ∈ R � � � � � � � � � log x � � Λ( n ) f ( n ) g ( n ) e( ϑn ) � ≪ x exp − c � � log log x � n � x for some positive constant c . The proof uses Schlickewei’s p -adic subspace theorem and Bakers’s theorem on linear form of logarithms. This does not permit to save a power of x . 21