Generalizations of a cotangent sum associated to the Estermann zeta function

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 µ on 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.

On a Category of Cotangent Sums Related to the Nyman-Beurling Criterion for the Riemann Hypothesis

“…The difference between the estimates from Theorems 2.1 and 2.2 is that stronger approximation techniques were applied in [19] to obtain more information about S(L; b). Namely, the generalized Euler summation formula (23) was used to improve the result of Theorem 2.1.…”

“…where ζ(s, x) is the Hurwitz zeta function. In [19] H. Maier and M. Th. Rassias tried to explain this phenomenon and obtained an important result, which establishes the equidistribution of certain normalized cotangent sums with respect to a positive measure.…”

“…Outline of the proof. In [19] H. Maier and M. Th. Rassias proved Theorem 2.2, which constitutes an improvement of their earlier Theorem 2.1.…”

The order of magnitude for moments for certain cotangent sums

“…The authors in a joint work [7] and the second author in his thesis, investigated the distribution of cotangent sums…”

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We settle a question on the rate of growth of the moments of cotangent sums considered by the authors in their previous papers [7,8].

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