Ergodic aspects of modern dynamics. Prime numbers in two bases

Transcription

1 Ergodic aspects of modern dynamics in honour of Mariusz Lemańczyk on his 60th birthday Bedlewo, 13 June 2018 Prime numbers in two bases Christian MAUDUIT Institut de Mathématiques de Marseille UMR 7373 CNRS, Université Aix-Marseille & Institut Universitaire de France. This talk is based on a joint work with Michael Drmota and Joel Rivat. 1

2 Prime numbers The prime numbers constitute a fascinating sequence which poses many difficult questions. Let us mention some known results and some open problems : Known prime numbers primes of the form an + b (Dirichlet theorem) ; primes such that αp (mod 1) belongs to some prescribed interval I [0, 1], for α R Q (Vinogradov-Davenport theorem) ; primes of the form [n c ] where 1 < c < c (Piatetski-Shapiro theorem) ; primes of the form a 2 + b 4 (Friedlander-Iwaniec theorem) ; primes of the form a 3 + 2b 3 (Heath-Brown theorem) ; arbitrarily long arithmetic progressions of primes (Green-Tao theorem). 2

3 Prime numbers Open problems are there infinitely many primes p such that p + 2 is a prime number (prime twins)? are there infinitely many primes p such that 2p + 1 is a prime number (Sophie Germain primes)? are there infinitely many primes of the form n 2 + 1? are there infinitely many primes of the form n! 1 or n! + 1 (factorial primes)? are there infinitely many primes of the form 2 n 1 (Mersenne primes, i.e. primes with no digit 0 in their representation in base 2)? are there infinitely many primes of the form 2 2n + 1 (Fermat primes, i.e. primes with exactly two digits 1 in their representation in base 2)? 3

4 Resolution of the Gelfond problem for prime numbers For any integer q 2 we denote by s q the sum of digits function in base q, defined by s q (n) = lj=0 n j for any positive integer n such that rep q (n) = n l... n 0. For any positive real number x and integers a and d, we denote by π(x; a, d) the number of primes p such that p a mod d. The following theorem answers a question asked in 1968 by Gelfond concerning the sum of digits of prime numbers. Theorem 1. (Mauduit-Rivat, 2010) For any positive integer m, there exists σ q (m) such that for any a Z we have card{p x, s q (p) a mod m} = (m, q 1) π(x; a, (m, q 1)) + O(x 1 σq(m) ). m 4

5 Prime numbers with an average sum of digit, Theorem 2. (Drmota-Mauduit-Rivat, 2009) The number of primes whose binary representation contains n digits 0 and n digits 1 is asymptotically equal to 4 n 1 π log 2n 3/2. Corollary 1. For any big enough integer k, there exists a prime which is the sum of k distinct powers of 2. Problem 1. Is this true for any integer k 1? Problem 2. Find an asymptotic estimate of the number of primes whose binary representation contains 2n digits 0 and n digits 1. 5

6 Prime numbers with preassigned digits Bourgain gave an asymptotic for the number of prime numbers with some preassigned digits, improving previous results due to Bourgain (2013), Harman-Kátai (2008), Harman (2006), Wolke (2005) and Kátai (1986) : Theorem 3. (Bourgain, 2015) For any given integer t > 0, for any 0 < j 1 < < j t < k 1 and for any (b 1,..., b t ) {0, 1} t, there exists c > 0 such that for any integer t verifying we have, for N = 2 k, k, card{p < N, p = k 1 j=0 1 t ck, Problem 3. Does it remain true for any c < 1? p j 2 j, (p j1,..., p jt ) = (b 1,..., b t )} 1 2 t N log N. 6

7 Prime numbers with missing digits Theorem 4. (Maynard, 2016) For any d {0,..., 9}, there are infinitely many prime numbers without digit d in their decimal representation. Problem 4. Find an estimate for the number of primes p x such that rep 3 (p) {0, 1}. 7

8 What do we want to do? Let q 1 and q 2 be positive coprime integers. Our goal is to find an estimate for the number of primes p x such that rep q1 (p) and rep q2 (p) satisfy some given properties. 8

9 q-additive and q-multiplicative functions Definition 1. A function h : N R is q-additive (resp. strongly q-additive) if for all (a, b) N {0,..., q 1}, we have (resp. h(aq + b) = h(a) + h(b)). h(aq + b) = h(aq) + h(b) A function f : N U is q-multiplicative (resp. strongly q-multiplicative) if for all (a, b) N {0,..., q 1}, we have (resp. f(aq + b) = f(a) f(b)). f(aq + b) = f(aq) f(b) Definition 2. A strongly q-multiplicative function is called proper if it is not of the form f(n) = e(ϑn) with (q 1)ϑ Z. 9

10 Caracteristic integer of a strongly q-additive function Definition 3. If h is a strongly q-additive integer valued function such that gcd (h(1),..., h(q 1)) = 1, the characteristic integer of h is d h = gcd (h(2) 2h(1),..., h(q 1) (q 1)h(1), q 1). If h is a strongly q-additive integer valued function such that gcd (h(1),..., h(q 1)) = 1, then f = e(αh) is proper if and only if d h α / Z. 10

11 Prime numbers along q-additive function Theorem 5. (Martin-Mauduit-Rivat, 2015) If h is a strongly q-additive integer valued function such that then for all (α, β) R 2 and x 2 we have n x gcd (h(1),..., h(q 1)) = 1, Λ(n) e ( αh(n) + βn ) (log x) 4 x 1 c q(h) d h α 2, where c q (h) is an explicit constant and the implicit constant depends only on q. Remark 2. If h = 0, then the best known upper bound (without the Riemann Hypothesis) is only of the size x exp ( c log x ) for some positive constant c. 11

12 Statistical independence of q-additive functions in different bases By using a general method concerning pseudorandom sequences in the sense of Bertrandias and generalizing previous results obtained by Mendès France, Bésineau showed in 1972 that, if q 1,...,q l are pairwise coprime bases and a 1,...,a l, m 1,..., m l are integers such that gcd(m i, q i 1) = 1 for any i {1,..., l}, then x card {n x, i {1,..., l}, s qi (n) a i mod m i } = + o(x). m 1 m l Kamae obtained in 1977 similar results when l = 2 by studying the mutual singularity of the spectral measures associated to the sum-of-digits functions. These results were extended by Queffelec in 1979 to multiplicatively independent bases and by Mauduit-Mosse in 1991 to different bases by using a slightly different method involving the study of some class of Riesz products. Grabner, Liardet and Tichy generalized in 2005 Kamae s result to l 2 q-additive functions in pairwise coprime bases by using ergodic methods. 12

13 Möbius orthogonality? Problem 5. Prove by ergodic methods that if, for any i {1,..., l} f i is a q i -multiplicative function, then we have n x µ(n)f 1 (n)... f l (n) = o(x) Remark 3. For l = 1, this has been done very recently by Konieczni, who told me that it can also be deduced by combining previous results due to Kátai 1986 and Indlekofer-Kátai This question is also related to several works by Lemańczyk concerning the study of ergodic properties of generalized Thue-Morse sequences. 13

14 Main theorem Theorem 6. (Drmota-Mauduit-Rivat, 2018) If f 1 is a strongly q 1 -multiplicative function and f 2 a strongly q 2 -multiplicative function such that gcd(q 1, q 2 ) = 1 and f 1 or f 2 is proper, then we have uniformly for ϑ R ( ) log x Λ(n)f 1 (n)f 2 (n) e(ϑn) x exp c n x log log x for some positive constant c. Remark 4. If f 1 and f 2 are not proper then the sum above is of the kind n x Λ(n) e(ϑ n) for some ϑ R for which the best known upper bound (without the Riemann Hypothesis) is only of the size x exp ( c log x ) for some positive constant c. 14

15 Prime numbers with a digit property in two bases Corollary 5. If h 1 is an integer valued strongly q 1 -additive function and h 2 is an integer valued strongly q 2 -additive function such that and gcd(q 1, q 2 ) = 1, gcd(h 1 (1),..., h 1 (q 1 1)) = 1 gcd(h 2 (1),..., h 2 (q 2 1)) = 1, then for any positive integers m 1, m 2 such that we have for all integers a 1, a 2, gcd(d h1, m 1 ) = gcd(d h2, m 2 ) = 1 card{p x, h 1 (p) a 1 mod m 1, h 2 (p) a 2 mod m 2 } = π(x) m 1 m 2 + o(π(x)). 15

16 Möbius orthogonality By the same method we can show that the product of two strongly q-multiplicative functions in coprime bases is orthogonal to the Möbius function. Theorem 7. (Drmota-Mauduit-Rivat, 2018) If f is a strongly q 1 -multiplicative function and g a strongly q 2 -multiplicative function such that gcd(q 1, q 2 ) = 1 and f or g is proper, then we have uniformly for ϑ R ( ) log x µ(n)f(n)g(n) e(ϑn) x exp c n x log log x for some positive constant c. The sequence (f(n)g(n)) n N in Theorem 7 is produced by a zero entropy dynamical system, so that this result can be seen as a new class of sequences verifying Möbius orthogonality in connection with the Sarnak conjecture. Remark 6. In the case of only one base, Theorem 7 is both a consequence from Theorem 5 and a consequence from a beautiful recent result due to Müllner saying that any q-automatic function f is orthogonal to the Möbius function. 16

17 In order to estimate sums of the form F (p) or Sum over prime numbers Λ(n)F (n) or p x n x n x µ(n)f (n) by using a combinatorial identity like Vaughan s identity. it is sufficient to estimate bilinear sums of the form m n a m b n F (mn). A classical process (Vinogradov, Vaughan, Heath-Brown) remains (using some more technical details), for some 0 < β 1 < 1/3 and 1/2 < β 2 < 1, to estimate uniformly the sums S I := m M n N F (mn) for M x β 1 (type I), where MN = x (which implies that the easy sum over n is long) and for all complex numbers a m, b n with a m 1, b n 1 the sums S II := m M n N (which implies that both sums have a significant length). a m b n F (mn) for x β 1 < M x β 2 (type II) 17

18 By Cauchy-Schwarz inequality : Sums of type II - Smoothing the sums S II 2 M m M n N b n F (mn) 2. Here, expanding the square and exchanging the summations, we would get a smooth sum over m, but also two free variables n 1 and n 2. However, we can get a useful control by using van der Corput s inequality : for z 1,..., z L C and R N we have 2 z l 1 l L L + R 1 R r <R ( 1 r R ) 1 l L 1 l+r L z l+r z l. The interest of this inequality is that now we have n 1 = n + r and n 2 = n so that the size of n 1 n 2 = r is under control. Now in fact we can take M = q µ, N = q ν and R = q ρ where µ, ν and ρ are integers such that ρ/(µ + ν) is very small. It remains to estimate non trivially b n+r b n F (m(n + r))f (mn). q ν 1 <n q ν q µ 1 <m q µ 18

19 Strategy of the proof The study of type I sums leads (by carry properties) to consider periodic arithmetic functions with period q λ 1 1 qλ 2 2. The first difficulty is to separate the contribution of the two bases and to combine arguments similar to the case of one base with new Diophantine and Fourier arguments. A second difficulty arises in the study of type II sums : the separation of the contributions coming from these two bases (by van der Corput and Cauchy-Schwarz inequalities) leads to much more difficult Fourier estimates than in the case of one base. We use estimates on average of the Fourier transform of correlations of strongly q-multiplicative functions (similar to U(2) Gowers norm) that are provided by combinatorial arguments. 19

20 Fourier Transforms of strongly q-multiplicative functions For any function ψ : Z C and any (a, b) Z 2 we denote by ψ [a] the function defined by n Z, ψ [a] (n) = ψ(n)ψ(n + a) and by ψ [a,b] the function defined by n Z, ψ [a,b] (n) = ψ(n)ψ(n + a)ψ(n + b)ψ(n + a + b) For any function f : N C and any λ N, let us denote by f λ the q λ -periodic function defined by n {0,..., q λ 1}, k Z, f λ (n + kq λ ) = f(n). The discrete Fourier transform of f λ is defined for t R by f λ (t) = 1 q λ 0 u<q λ f λ (u) e ( ut q λ ). 20

21 Estimates on average of the Fourier transforms The following estimates are crucial in the proof of Theorem 6 : Proposition 7. If f is a proper strongly q-multiplicative function, then there exist constants c 1 > 0, c 2 > 0 such that for all λ N we have 1 q λ f [b] 0 b<q λ 0 h<q λ λ (h) 4 c 1 q c 2λ. Proposition 8. If f 0 is an integer valued strongly q-additive function such that gcd(f 0 (1),..., f 0 (q 1 1)) = 1 and f = e(αf 0 ), then there exist constants c 1 > 0, c 2 > 0 such that uniformly for α R such that d f0 α / Z we have 1 q 2λ 0 a<q λ f [a,b] 0 b<q λ λ (t) 2 c 1 q c 2 d f0 α 2 / log(1/ d f0 α ) λ. 21

22 A typical combinatorial argument in the estimates of the Fourier transforms Lemme 9. Suppose that q 2. Then for every strongly q-multiplicative function f there exist L 2, a proper subset S {0, 1,..., q 1} L and a constant c = c (q, f, L, S) > 0 such that f [b] λ (h) q c J(h), where J(h) denotes the number of sub-blocks of length L in the q-ary expansion of h = λ 1 j=0 ε j(h) that are not contained in S. 22

23 A Diophantine argument in the estimate of the sums of type II A last difficulty appears in the non-diagonal terms of the sums of type II which leads to estimate a linear form of logarithms. More precisely, we need to have lower bounds for quantities of the kind with (h 1, h 2 ) Z 2 and (µ 1, µ 2 ) N 2. h 1 h 2 q µ 1 1 q µ By Baker s theorem, we get lower bounds of the kind h 1 h 2 q µ 1 1 q µ exp ( C log q 1 log q 2 log max(µ 1, µ 2 ) log max( h 1, h 2 )) for some constant C > 0, which finally allows us to win a factor of the size for some constant C > 0. exp(c log x/ log log x) 23

24 .