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

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.

References in corpus (2)

• Estimating $\pi (x)$ and related functions under partial RH assumptions
• A bound for the error term in the Brent-McMillan algorithm