Generalizations of a cotangent sum associated to the Estermann zeta function

TL;DR

This paper investigates the distribution and asymptotic behavior of cotangent sums linked to the Estermann zeta function, providing new generalizations, moment estimates, and a measure for their equidistribution, relevant to the Riemann Hypothesis.

Contribution

It proves the existence of a unique measure for the equidistribution of normalized cotangent sums and extends asymptotic formulas and moment estimates for these sums.

Findings

Existence of a unique positive measure for cotangent sum distribution

Asymptotic formulas for generalized cotangent sums

Growth rate estimates for moments of cotangent sums

Abstract

Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure {\mu} on $\mathbb{R}$, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k.