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

Details Sums of the error term function in the mean square for $ζ(s)$ Sums of the error term function in the mean square for $ζ(s)$

2007-07-29

arxiv.org/abs/0707.4275v1

Monats. Mathematik, 155(2008), 107-118

Mathematics

Number Theory

14 pages

Scientific paper

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.

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