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arXiv · 0707.4275

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

Yann Bugeaud·Aleksandar Ivić
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Abstract

Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|ζ(1/2+it)|^2 dt - T\Bigl(\log {T\over2π} + 2γ-1\Bigr)$$ is the error term in the mean square formula for $|ζ(1/2+it)|$. 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 ${\rm e}^{2πm}$ and for the partial quotients in its continued fraction expansion.

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BibTeXRIS

Yann Bugeaud, Aleksandar Ivić. 2007-07-29. Sums of the error term function in the mean square for $ζ(s)$. https://arxiv.org/abs/0707.4275

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