Mean values of ratios of the Riemann zeta function

Abstract

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

arXiv comment10 pages; incorporated referee comments; added two new conjectures