Lower bounds for discrete moments in a conjecture of Ng

Abstract

Assuming the Riemann Hypothesis and the simplicity of the non-trivial zeros of the Riemann zeta function $\zeta (s)$, we prove lower bounds for the discrete moments ${\sum }_{0<\gamma \le T}|\zeta (2\rho )/{\zeta }^{\prime }(\rho ){|}^{2k}$ for every fixed rational $k\ge 1$, and for every fixed real $k\ge 1$. At $k=1$ we give a new proof of the explicit bound, half of the main term conjectured by Ng.

arXiv commentv2 is substantially expanded and has a new title (v1: "A note on a conjecture of Ng"). New in v2: lower bounds of the form $T(logT{)}^{(k-1{)}^2}$. The $k=1$ bound of v1 was obtained earlier in the 2025 PhD thesis of S. Sinha; v2 acknowledges this, and notes that the proof here, found independently, differs from Sinha's