Mean values of ratios of the Riemann zeta function
Acta Arithmetica
MSC: Primary 11M06
DOI: 10.4064/aa230720-10-6
Published online: 29 October 2024
Abstract
We prove that $$\int _{T}^{2T} \bigg|\frac {\zeta ({1}/{2}+{\rm i} t)}{\zeta (1+2{\rm i} t)}\bigg |^2\, {\rm d} t = \frac {1}{\zeta (2)} T \log T + \biggl ( \frac {\log \frac {2}{\pi } + 2\gamma -1 }{\zeta (2)} -4 \,\frac {\zeta ^{\prime }(2)}{\zeta ^2(2)} \biggr ) T + O(T\, (\log T)^{-2023} )$$ for all $T \geqslant 100$. For given $a\in \mathbb N $, we also establish similar formulas for second moments of $|\zeta (1/2 + {\rm i} t)/\zeta (1 + {\rm i} at)|$, namely $$\lim _{a \to \infty } \lim _{T \to \infty }\frac {1}{T \log T} \int _{T}^{2T} \bigg|\frac {\zeta ({1}/{2}+{\rm i} t)}{\zeta (1+{\rm i} at)}\bigg |^2\, {\rm d} t = \frac {\zeta (2)}{\zeta (4)}. $$
