Mean values of ratios of the Riemann zeta function

TL;DR

This paper derives an asymptotic formula for the mean square of the ratio of the Riemann zeta function on the critical line to its value at shifted points, revealing new insights into their average behavior.

Contribution

It provides a precise asymptotic for the second moment of zeta ratios and establishes a limit relation involving these ratios as the shift parameter grows.

Findings

Asymptotic formula for the mean square of zeta ratio involving log T

Limit of the ratio of second moments as shift parameter tends to infinity

Explicit constants involving zeta function values in the asymptotic expression

Abstract

It is proved that $$\int_{T}^{2T} \left|\frac{\zeta\left(\frac{1}{2}+{\rm i} t\right)}{\zeta\left(1+2{\rm i} t\right)}\right|^2 {\rm d} t = \frac{1}{\zeta(2)} T \log T + \left( \frac{\log \frac{2}{\pi} + 2\gamma -1 }{\zeta(2)} -4 \,\frac{\zeta^{\prime}(2)}{\zeta^2(2)} \right) T + O\left(T\, \left(\log T\right)^{-2023} \right) , \quad \forall T \geqslant 100. $$ For given $a \in \mathbb N$, we also establish similar formulas for second moments of $|\zeta(\tfrac{1}{2} + {\rm i} t)/\zeta(1 + {\rm i} at)|.$ We have \begin{align*} \lim_{a \to \infty} \lim_{T \to \infty}\frac{1}{T \log T} \int_{T}^{2T} \left|\frac{\zeta\left(\frac{1}{2}+{\rm i} t\right)}{\zeta\left(1+{\rm i} at\right)}\right|^2 {\rm d} t = \frac{\zeta(2)}{\zeta(4)}. \end{align*}