On the correlation of shifted values of the Riemann zeta function

TL;DR

This paper extends Soundararajan's work on the Riemann zeta function to include shifted moments, providing bounds and conjectures on their asymptotic behavior, and analyzing the correlation transition based on shifts.

Contribution

It generalizes methods to shifted moments of the zeta function, establishes bounds, and proposes conjectures on their asymptotics using Random matrix theory.

Findings

Correlation of shifted zeta values transitions at shift differences around 1/ log T.

Shifted moments exhibit near independence when shifts are much larger than 1/ log T.

Provides upper and lower bounds for shifted moments and conjectures asymptotic formulas.

Abstract

In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment} proved that $\int_{0}^T |\zeta(1/2 + it)|^{2k} dt \ll_{k, \epsilon} T(\log T)^{k^2 + \epsilon}$ for every $k$ positive real number and every $\epsilon > 0.$ In this paper we 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 $|\zeta(\h + it + i\alpha_1)|$ and $|\zeta(\h + it + i\alpha_2)|$ transition at $|\alpha_1 - \alpha_2| \approx \frac{1}{\log T}$. In particular these distribution appear independent when $|\alpha_1 - \alpha_2|$ is much larger than $\frac{1}{\log T}.$