Paper detail

Generalizations of a cotangent sum associated to the Estermann zeta function

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 μ 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.