ON THE CORRELATION OF SHIFTED VALUES OF THE RIEMANN

arXiv:0910.0664v1 [math.NT] 5 Oct 2009

ON THE CORRELATION OF SHIFTED VALUES OF THE RIEMANN ZETA FUNCTION VORRAPAN CHANDEE Abstract. In 2007, assuming the Riemann Hypothesis (RH), Soundararajan R 2 [11] proved that 0T |ζ(1/2 + it)|2k dt ≪k,ǫ T (log T )k +ǫ for every k positive real number and every ǫ > 0. In this paper I generalized his methods to find upper bounds for shifted moments. We also obtained their lower bounds and conjectured asymptotic formulas based on Random matrix model, which is analogous to Keating and Snaith’s work. These upper and lower bounds suggest that the correlation of |ζ( 21 + it + iα1 )| and |ζ( 21 + it + iα2 )| transition at |α1 −α2 | ≈ log1 T . In particular these distribution appear independent when |α1 − α2 | is much larger than

1 log T

.

1. Introduction Finding moments of the Riemman zeta function ζ(s) is an important problem in analytic number theory, especially the moments on the critical line: Z T 1 ζ( + it) 2k dt. Mk (T ) := 2 0

Extensive work has been done to find an asymptotic formula for Mk (T ); however, the only unconditional results in this direction are proven for k = 1, due to Hardy and Littlewood, and k = 2, due to Ingham [14]. Assuming the Riemann hypothesis (RH), good upper and lower bounds are available. Ramachandra [8] proved that 2 for any positive real integer k, Mk (T ) ≫ T logk T. Later in 2007, Soundararajan [11] showed that for every positive real number k and every ǫ > 0 (1)

Mk (T ) ≪k,ǫ T (log T )k

2

+ǫ

.

In 2000, Keating and Snaith [4] conjectured an asymptotic formula for Mk (T ), for every positive integer k, based on the random matrix model for the zeros of ζ(s). They suggested that the value distribution of ζ(1/2 + it) is related to that Q of the characteristic polynomials of random unitary matrices, Λ(eiθ ) := N n=1 (1 − i(θn −θ) e ). Therefore they computed the moments of the characteristic polynomials to arrive at a conjecture for Mk (T ) and showed that Z G2 (k + 1) k2 N , |Λ(eiθ )|2k dUN ∼ (2) gU (N, k) := G(2k + 1) U(N ) Date: October 5, 2009. 1

2

VORRAPAN CHANDEE

T , this led them to where G is the Barnes G-function. Using the scaling N = log 2π 2

2

(k+1) k T, where conjecture that Mk (T ) ∼ a(k) G G(2k+1) T log

(3)

a(k) :=

Y p

1−


2 1 k p

∞ X

m=0

Γ(m+k)
2 −m
. p m!Γ(k)

This conjecture agrees with the known results for k = 1, 2.

A generalization of the moments of ζ(s) are the shifted moments, defined as Z T → − (4) Mk (T, α ) = |ζ( 12 + it+ iα1 )|2k1 |ζ( 12 + it+ iα2 )|2k2 ...|ζ( 12 + it+ iαm )|2km dt, 0

→ where k = (k1 , k2 , ..., km ) is a sequence of positive real numbers and − α = (α1 , ...αm ), where αi 6= αj when i 6= j, |αi − αj | = O(1), and αi = O(log T ). Also αi = αi (T ) is a real valued function in terms of T such that limT →∞ αi log T and limT →∞ (αi − αj ) log T exists or equals ±∞. Conrey, Farmer, Keating, Rubinstein and Snaith [1] gave a general recipe from which an asymptotic formula for the shifted moments of the Riemann zeta function may be conjectured. However, it is not immediately clear from their recipe what the leading asymptotic term for these shifted moments should be, and this is elucidated by K¨ osters in [6]. Specifically, based on the work in [1], K¨osters conjectures that for any T0 > 1 and µ1 , ..., µM ∈ R, Z T 1 2πµ1 2 2πµM 2 1 (5) |ζ( 12 + it + i log lim 2 T )| ...|ζ( 2 + it + i log T )| dt T →∞ T (log T )M T0 a(M ) det(bjk )j,k=1,...,M , = 2 ∆ (2πµ1 , ..., 2πµM ) Q where ∆(x1 , .., xn ) = 1≤j