Mean values of ratios of the Riemann zeta function

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*}