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

Abstract

Sums of the form ${\sum }_{n\le x}{E}^k(n)(k\in \mathbf{N}$ fixed) are investigated, where $$E(T)={\int }_0^T|\zeta (1/2+it){|}^{2}dt-T(\log \frac{T}{2\pi }+2\gamma -1)$$ 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 ${\mathrm{e}}^{2\pi m}$ and for the partial quotients in its continued fraction expansion.

14 pages