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arXiv · 0910.0664

On the correlation of shifted values of the Riemann zeta function

Vorrapan Chandee
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Abstract

In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment} proved that $\int_{0}^T |ζ(1/2 + it)|^{2k} dt \ll_{k, ε} T(\log T)^{k^2 + ε}$ for every $k$ positive real number and every $ε> 0.$ 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 $|α_1 - α_2| \approx \frac{1}{\log T}$. In particular these distribution appear independent when $|α_1 - α_2|$ is much larger than $\frac{1}{\log T}.$

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Vorrapan Chandee. 2009-10-05. On the correlation of shifted values of the Riemann zeta function. https://arxiv.org/abs/0910.0664

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