Moments of the Riemann Zeta Function and Random Matrix Theory. Chris Hughes

Transcription

1 Moments of the Riemann Zeta Function and Random Matrix Theory Chris Hughes

2 Overview We will use the characteristic polynomial of a random unitary matrix to model the Riemann zeta function. Using this, we will present a conjecture for the moments of the zeta function. We will use this conjecture to predict extremely small values of the zeta function. This will naturally lead into the study of discrete moments of the derivative of zeta. We will use random matrix theory to make a conjecture about such moments.

3 For Re(s) > The Riemann zeta function ζ(s) = n= n s = p prime p s. Analytic extension into the whole complex plane, apart from a simple pole at s = Functional equation: ζ(s) = χ(s)ζ( s) where χ(s) = π s /2 Γ( 2 2 s) Γ( 2 s)

4 non-trivial zeros, s = /2 + iγn The zeros and poles of the Riemann zeta function (Not to scale) trivial zeros, s = 2n simple pole at s = the critical line, Re(s) = /2 Number of zeros in critical strip: N(T ) = T log T e + O(log T ) If ρ is a zero, so is ρ, ρ, ρ. Riemann Hypothesis: Re(ρ) = 2 Under RH, ρ = ρ.

5 Random unitary matrices Let U(N) be the set of N N unitary matrices. It is a compact Lie group. This means it has a (unique) Haar measure (leftand right-invariant measure). That is for any measurable set A U(N) and any U U(N), P Haar {A} = P Haar {UA} = P Haar {AU} H. Weyl calculated the probability density of eigenangles under Haar measure. It equals () N N! j<k N e iθ j e iθ k where θ,..., θ N are the eigenvalues. 2

6 Take an N N unitary matrix, and scale its eigenangles by N, so they have average spacing unity. Scale the non-trivial zeros around height T by log T, so they have average spacing unity. There is much evidence (theoretical, heuristic and numerical) relating statistics of the scaled zeros of the Riemann zeta function high up the critical line, with statistics of the scaled eigenangles of large Haar-measured unitary matrices.

7 This led Keating and Snaith to model the Riemann zeta function with the characteristic polynomial, Z(U, θ) := det(i Ue iθ ) = N ( e i(θn θ) ) n= of an N N random unitary matrix, U, with eigenvalues e iθ n. Equate density of eigenangles with density of zeros: N = log T

8 Central limit theorems Theorem (Selberg). For rectangles B C, T meas T t 2T : log ζ( 2 + it) B 2 log log T B e (x2 +y 2 )/2 dx dy. Theorem (Keating and Snaith). For fixed θ, P Haar log Z(U, θ) B 2 log N B e (x2 +y 2 )/2 dx dy

9 Graph of the negative log of the value distribution of log ζ( 2 + it) around the 020 th zero (red), against the negative log of the probability density of log Z(U, 0) with N = 42 (green).

10 Set σ := 2 log N. One can actually prove that log Z(U, θ)/σ is ergodic, in the sense that for almost all U the distribution of log Z(U, θ)/σ over θ is the same as the distribution of log Z(U, θ)/σ over U, as N. Theorem (CPH, Keating and O Connell). Denote by m the uniform probability measure on ( π, π] (so that m(dθ) = dθ/). The sequence of laws m log Z(U, θ) 2 log N converges weakly in probability to X + iy, where X, Y are iid standard normal random variables.

11 The moments of the zeta function Theorem (Baker and Forrester, Keating and Snaith). For any fixed θ, E { Z(U, θ) 2k} = N j= Γ(j)Γ(j + 2k) Γ(j + k) 2 G2 (k + ) G(2k + ) N k 2 G is the Barnes G function, and satisfies G(z + ) = Γ(z)G(z). Some results: k E{ Z(U, θ) 2k } N 2 2 N ! N ! N 6 as N

12 There is a folklore conjecture that T where T 0 a(k) = p prime ζ( 2 + it) 2k dt g k a(k) ( ) k 2 p m=0 ( log T ) k 2 ( ) 2 Γ(m + k) p m m! Γ(k) k T T 0 ζ( 2 + it) 2k dt log T 2 2 a(2)(log T ) ! a(3)(log T ) ! a(4)(log T )6

13 The Keating-Snaith conjecture Since N = log T, this leads to Conjecture (Keating and Snaith). T T 0 ζ( 2 + it) 2k dt ( G2 (k + ) G(2k + ) a(k) log T ) k 2 as T.

14 Very small values of ζ( 2 + it) Theorem (CPH, Keating, O Connell). As N, if 0 < y < N (/2+ɛ), Proof. P { Z(U, 0) y} G 2 ( 2 )N /4 y. where P { Z(U, 0) y} = p(t) = e ixt E log y p(t) dt { Z(U, 0) ix} dx When t < ( + ɛ) log N, the Fourier integral is 2 dominated by the pole at x = i, and so, { p(t) i Res e ixt E { Z(U, 0) ix}} x=i e t G 2 ( /4 )N 2

15 Define P (T, y) to equal T meas { 0 t T : ζ( 2 + it) y}. A similar calculation using the Keating-Snaith conjecture for the moments of the zeta functions leads to: Conjecture. If y < (log T ) (/2+ɛ), then as T. ( P (T, y) G 2 ( 2 )a( 2 )y log T ) /4 From the graph: T = and N = 42. log P (T, e 0 ) + log P {log Z(0) 0} whereas log a( 2 ) =

16 As y 0+, P (T, y) 2y T 0<γ n T ζ ( 2 + iγ n) which leads to Conjecture. N(T ) 0<γ n T ζ ( 2 + iγ n) ( πg 2 ( 2 )a( 2 ) log T ) 3/4 It is natural then to ask Question. Can random matrix theory model N(T ) 0<γ n T ζ ( 2 + iγ n) 2k?

17 Discrete moments of the derivative We wish to study { } N E Z (U, θ n ) 2k = E { Z (U, θ ) 2k} N n= by rotational invariance of Haar measure. Theorem (CPH, Keating, O Connell). E { Z (U, θ ) 2k} G2 (k + 2) G(2k + 3) N k(k+2)

18 Conjecture. N(T ) 0<γ n T ζ ( 2 + iγ n) 2k G2 (k + 2) G(2k + 3) a(k) Agrees with all known results: Theorem (Gonek). Under RH, N(T ) 0<γ n T ζ ( 2 + iγ n) 2 2 ( log T ) k(k+2) ( log T ) 3 Conjecture (Gonek). If all the zeros are simple and RH is true, then ζ ( N(T ) 2 + iγ n) 2 6 ( π 2 log T ) 0<γ n T

19 Taking a theorem due to Conrey, Ghosh and Gonek, and extending it beyond the range of proven applicability, one can deduce Conjecture. N(T ) 0<γ n T ζ ( 2 + iγ n) 4 440π 2 ( log T ) 8 This is of the correct order of magnitude: Theorem (Nathan Ng). ( 97 6) π 2 N(T ) ( log T ) 8 0<γ n T Numerically, the bounds are ζ ( 2 + iγ n) 4 ( ) π ( log T ) 8

20 Theorem (Hejhal). Under RH and an assumption that the zeros don t get too close together, log ζ (/2+iγ n ) log T 2 log log T when averaged over all zeros between T and 2T, behaves like a standard normal random variable. Similarly, Theorem (CPH, Keating, O Connell). log Z (U, θ ) log N 2 log N converges in distribution to a standard normal random variable.

21 Summary Statistically zeros of zeta behave like eigenangles of a random unitary matrix. One can model zeta by the characteristic polynomial of a unitary matrix. The model works well in the local regime (central limit theorems). The model must be corrected by a non-random matrix (zeta-specific) term in the global regime (moments / large deviations).