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

Helmut Maier () and Michael Th. Rassias ()
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Helmut Maier: University of Ulm, Department of Mathematics
Michael Th. Rassias: Princeton University, Department of Mathematics

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 277-293 from Springer

Abstract: 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.

Keywords: Cotangent sums; Dirichlet series; Equidistribution; Estermann zeta function; Fractional part; Moments; Riemann hypothesis; Riemann zeta function; Primary 26A12; Secondary 11L03, 11M06 (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-28203-9_18

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DOI: 10.1007/978-3-319-28203-9_18

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