Distribution of a cotangent sum related to the Nyman–Beurling criterion for the Riemann Hypothesis

摘要

A certain category of cotangent sums has been proven of importance in the Nyman–Beurling criterion for the Riemann Hypothesis. In previous work ([12], [13]) the authors proved the existence of a unique positive measure μ on R, with respect to which certain normalized cotangent sums, evaluated at rational numbers with fixed denominators are equidistributed. The tools applied in this paper belong to various fields of Mathematics, for instance the relation between the equidistribution mod1 of the multiples of a number and the Diophantine approximation properties of that number. In this paper we prove an analogous result for the case that the denominator of the rational numbers is a fixed prime number and that the numerator is also prime.