Sums of the error term function in the mean square for $\zeta(s)$

TL;DR

This paper investigates sums involving powers of the error term in the mean square formula for the Riemann zeta function, focusing on the case k=1, and employs bounds related to irrationality measures and continued fractions.

Contribution

It introduces new bounds and analysis for sums of the error term in the mean square formula of the zeta function, especially for the case k=1, using irrationality measure techniques.

Findings

Derived bounds for sums of the error term for k=1

Compared difficulty with the divisor problem case

Utilized irrationality measures and continued fractions

Abstract

Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|\zeta(1/2+it)|^2 dt - T\Bigl(\log {T\over2\pi} + 2\gamma -1\Bigr)$$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. The emphasis is on the case k=1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of ${\rm e}^{2\pi m}$ and for the partial quotients in its continued fraction expansion.