On the correlation of shifted values of the Riemann zeta function

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 $|ζ(\h + it + iα_1)|$ and $|ζ(\h + it + iα_2)|$ 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 $\frac{1}{\log T}.$

24 pages