Asymptotics and Equidistribution of Cotangent Sums Associated with the Estermann and Riemann Zeta Functions

Abstract

The Nyman–Beurling criterion is a well-known approach to the Riemann Hypothesis. Certain integrals over Dirichlet series appearing in this approach can be expressed in terms of cotangent sums. These cotangent sums are also associated with the Estermann zeta function. In this paper improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. The main result of this paper is the existence of a unique positive measure μ on \(\mathbb{R}\) with respect to which normalized versions of these cotangent sums are equidistributed. We also consider the moments of order 2k as a function of k.