Fourier optimization and prime gaps
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1 Fourier optimization and prime gaps Emanuel Carneiro IMPA - Rio de Janeiro (joint with M. Milinovich and K. Soundararajan) San Diego, Jan 2018 Emanuel Carneiro Fourier optimization San Diego, Jan / 14
2 A classical problem 1 For f : R R, our normalization for Fourier transform is f (t) = e 2πixt f (x) dx. 2 Let F : R R be a nonnegative function, with F(0) = 1 and supp( F) [ 1, 1]. What is the minimal value of F L 1 (R)? 3 Answer =1. F(x) = (sin(πx)/(πx)) 2 R Emanuel Carneiro Fourier optimization San Diego, Jan / 14
3 An innocent variant 1 Let F : R R be such that F(0) = 1 and supp( F) [ 1, 1]. What is the minimal value of F L 1 (R)? 2 H(x) = cos(2πx) 1 16x 2 yields H L 1 (R) = Best up-to-date (Gorbachev 05) < C < There exists a unique extremizer. Emanuel Carneiro Fourier optimization San Diego, Jan / 14
4 Prime gaps 1 Bertrand s postulate (1845): is there always a prime in the interval [x, 2x]? 2 Tchebyshev (1852): Yes. There is always a prime in [x, x + x log x ] for x large. 3 Hoheisel (1930): There is always a prime in [x, x + x θ ] for some 0 < θ < 1, and x large. 4 Baker - Harman - Pintz (2001): There is always a prime in [x, x + x ] for x large. Emanuel Carneiro Fourier optimization San Diego, Jan / 14
5 Prime gaps on RH Cramér s bounds (1920) 1 p n+1 p n = O( p n log p n ), i.e. every interval [x, x + c x log x], for some c > 0, contains a prime when x is large. 2 Historic progress: Goldston 83: c = 4. Ramaré and Saouter 03: c = 8/5 Dudek 15: c = 1 + o(1). 3 Non-asymptotic version - best result due to Dudek, Grenié, Molteni 16: for x 4, every interval [x, x + c x log x] contains a prime. Here c = log x. Ramaré and Saouter 03 (c = 8/5) Emanuel Carneiro Fourier optimization San Diego, Jan / 14
6 Recent progress joint with M. Milinovich and K. Soundararajan Theorem (Asymptotic version) Assume RH. For x large, every interval [ x, x + 21 ] x log x 25 contains a prime. Theorem (Non-asymptotic version) Assume RH. For x 4, every interval [ x, x + 22 ] x log x 25 contains a prime. Emanuel Carneiro Fourier optimization San Diego, Jan / 14
7 Strategy 1 Explicit formulas connecting zeros of ζ(s) and primes. 2 Brun-Titchmarsh inequality. 3 Fourier optimization problems. Emanuel Carneiro Fourier optimization San Diego, Jan / 14
8 Using the explicit formula Let h be a function with mild smoothness/decay properties: ( ) ρ 1 ( ) ( 2 1 h = h + h 1 ) 1 ĥ(0) log π i 2i 2i 2π ρ + 1 ( h(u) R Γ 1 2π Γ 4 + iu ) du 2 1 ( ) ( )) Λ(n) log n log n (ĥ + 2π n 2π ĥ, 2π n 2 where ρ = β + iγ are the non-trivial zeros of ζ(s). Assume that an interval [x, x + c x log x] has no primes. Let F be a test function. Idea is to use the formula with h(z) = F( z) a iz with [x, x + c x log x] = [a e 2π, a e 2π ]. Perform an asymptotic analysis (as 0 and a ). Emanuel Carneiro Fourier optimization San Diego, Jan / 14
9 Using the explicit formula Let h be a function with mild smoothness/decay properties: ( ) ρ 1 ( ) ( 2 1 h = h + h 1 ) 1 ĥ(0) log π i 2i 2i 2π ρ + 1 ( h(u) R Γ 1 2π Γ 4 + iu ) du 2 1 ( ) ( )) Λ(n) log n log n (ĥ + 2π n 2π ĥ, 2π n 2 where ρ = β + iγ are the non-trivial zeros of ζ(s). Assume that an interval [x, x + c x log x] has no primes. Let F be a test function. Idea is to use the formula with h(z) = F( z) a iz with [x, x + c x log x] = [a e 2π, a e 2π ]. Perform an asymptotic analysis (as 0 and a ). Emanuel Carneiro Fourier optimization San Diego, Jan / 14
10 This leads to... 1 If supp( F) [ 1, 1] we would have c F 1 F(0). 2 One can actually do better by (over)estimating in [ 1, 1] c : c F 1 (F(0) B [ 1,1] c ( F(t) )+ dt ). 3 Here B is the Brun-Titchmarsh constant in our desired scale B := lim sup x π(x + x) π(x) x/ log x. 4 By the PNT (on the left) and work of Iwaniec (on the right): 1 B Emanuel Carneiro Fourier optimization San Diego, Jan / 14
11 This leads to... 1 If supp( F) [ 1, 1] we would have c F 1 F(0). 2 One can actually do better by (over)estimating in [ 1, 1] c : c F 1 (F(0) B [ 1,1] c ( F(t) )+ dt ). 3 Here B is the Brun-Titchmarsh constant in our desired scale B := lim sup x π(x + x) π(x) x/ log x. 4 By the PNT (on the left) and work of Iwaniec (on the right): 1 B Emanuel Carneiro Fourier optimization San Diego, Jan / 14
12 This leads to... 1 If supp( F) [ 1, 1] we would have c F 1 F(0). 2 One can actually do better by (over)estimating in [ 1, 1] c : c F 1 (F(0) B [ 1,1] c ( F(t) )+ dt ). 3 Here B is the Brun-Titchmarsh constant in our desired scale B := lim sup x π(x + x) π(x) x/ log x. 4 By the PNT (on the left) and work of Iwaniec (on the right): 1 B Emanuel Carneiro Fourier optimization San Diego, Jan / 14
13 Extremal problems 1 Problem 1: Given 1 A, find 1 ( ) C(A) := sup F(0) A F(t) dt, F A F 1 [ 1,1] c F 0 where the supremum is taken over the class A of continuous functions F : R C, with F L 1 (R). 2 Problem 2: Given 1 A, find C + 1 ( ) ) (A) := sup F(0) A ( F(t) F A + F 1 [ 1,1] c + dt, F 0 where the supremum is taken over the class A + of even and continuous functions F : R R, with F L 1 (R). Emanuel Carneiro Fourier optimization San Diego, Jan / 14
14 What we can prove 1 Existence of extremizers for A > 1. No extremizers for A = 1. 2 Uniqueness in the bandlimited problem (+ variational condition). 3 Good upper and lower bounds for all of these problems. 4 A C(A) and A C + (A) are monotone decreasing with 2 = C + (1) = C(1) C + (A) C(A) C + ( ) C( ) ( ) 36 C + > = F(x) = 4.8 x 2 e 3.3x x 2 e 7.4x x 24 e 9.7x e 2.8x e 2x 2 Emanuel Carneiro Fourier optimization San Diego, Jan / 14
15 Conclusion 1 Theorem (Asymptotic version) Assume RH. Then lim sup n p n+1 p n 1 pn log p n C + (B) 1 C + (36/11) Emanuel Carneiro Fourier optimization San Diego, Jan / 14
16 Conclusion 2 - non-asymptotic version 1 Use version of the Brun-Titchmarsh inequality due to Montgomery and Vaughan. π(x + y) π(x) < 2y log y, for all x, y > 1. Relevant range is y x, which corresponds to A = 4 in the extremal problem. 2 With H(x) = cos(2πx) 1 16x 2 and λ = 0.9 we use F(x) = H(x/λ). Then J(F) = F(0) 4 [ 1,1] c F(t) dt F 1 = > Work the previous argument to make all error terms effective (Mellin transform approach slightly simpler). Emanuel Carneiro Fourier optimization San Diego, Jan / 14
17 ICM 2018 Emanuel Carneiro Fourier optimization San Diego, Jan / 14
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