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

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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.

Dedicated to the memory of Professor Wolfgang Schwarz

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Acknowledgements

We express our thanks to the referee for providing valuable remarks.

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Authors and Affiliations

  1. Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081, Ulm, Germany

    Helmut Maier

  2. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA

    Michael Th. Rassias

  3. Department of Mathematics, ETH-Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Michael Th. Rassias

Authors
  1. Helmut Maier
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  2. Michael Th. Rassias
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Correspondence to Michael Th. Rassias .

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Editors and Affiliations

  1. Institut für Mathematik und Angewandte Informatik, Stiftung Universität Hildesheim, Hildesheim, Germany

    Jürgen Sander

  2. Institut für Mathematik, Julius-Maximilians Universität Würzburg, Würzburg, Germany

    Jörn Steuding

  3. Institut für Mathematik, Julius-Maximilians Universität Würzburg, Würzburg, Germany

    Rasa Steuding

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Maier, H., Rassias, M.T. (2016). Asymptotics and Equidistribution of Cotangent Sums Associated with the Estermann and Riemann Zeta Functions. In: Sander, J., Steuding, J., Steuding, R. (eds) From Arithmetic to Zeta-Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-28203-9_18

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