A Harmonic Sum over Nontrivial Zeros of the Riemann Zeta-Function

Abstract

We consider the sum <![CDATA[ $\sum 1/\gamma $[]>, where <![CDATA[ $\gamma $[]> ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval <![CDATA[ $(0,T]$[]>, and examine its behaviour as <![CDATA[ $T \to \infty $[]>. We show that, after subtracting a smooth approximation <![CDATA[ $({1}/{4\pi }) \log ^2(T/2\pi),$[]> the sum tends to a limit <![CDATA[ $H \approx-0.0171594$[]>, which can be expressed as an integral. We calculate H to high accuracy, using a method which has error <![CDATA[ $O((\log T)/T^2)$[]>. Our results improve on earlier results by Hassani ['Explicit approximation of the sums over the imaginary part of the non-trivial zeros of the Riemann zeta function', Appl. Math. E-Notes 16 (2016), 109-116] and other authors.