Generalization of the Ford-Zaharescu Theorem

TL;DR

This paper extends the Ford-Zaharescu theorem by deriving an asymptotic formula for sums involving linear combinations of the imaginary parts of the Riemann zeta zeros, with a focus on a specific class of functions.

Contribution

It generalizes the Ford-Zaharescu theorem to broader classes of functions and sums over zeros with linear combinations where coefficients sum to zero.

Findings

Derived an asymptotic formula for the sum H.

Extended the theorem to a wider class of functions h.

Provided insights into the distribution of zeta zeros through these sums.

Abstract

We derive an asymptotic formula for the sum $$ H = \sum_{0<\gamma_k\leqslant T,\, 1\leqslant k\leqslant m}h(a_1\gamma_1+a_2\gamma_2+\cdots + a_m\gamma_m), $$ where $a_1, a_2, \ldots, a_m$ are integers whose sum equals zero, $\gamma_1, \ldots, \gamma_m$ independently run through the imaginary parts of the non-trivial zeros of the Riemann zeta function, each zero occuring in the sum the number of times of its multiplicity, and the function $h$ belongs to some special class.