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arXiv · 2009.05251

A harmonic sum over nontrivial zeros of the Riemann zeta-function

Richard P. Brent·David J. Platt·Timothy S. Trudgian
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Abstract

We consider the sum $\sum 1/\gamma$, where $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a smooth approximation $\frac{1}{4\pi} \log^2(T/2\pi),$ the sum tends to a limit $H \approx -0.0171594$ which can be expressed as an integral. We calculate $H$ to high accuracy, using a method which has error $O((\log T)/T^2)$. Our results improve on earlier results by Hassani and other authors.

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Richard P. Brent, David J. Platt, Timothy S. Trudgian. 2020-09-11. A harmonic sum over nontrivial zeros of the Riemann zeta-function. https://doi.org/10.1017/s0004972720001252

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