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Cotangent Sums Related to the Riemann Hypothesis for Various Shifts of the Argument

Part of: Classical measure theory Elementary classical functions Approximations and expansions Zeta and $L$-functions: analytic theory Distribution theory - Probability

Published online by Cambridge University Press:  15 October 2019

Helmut Maier  and
Michael Th. Rassias
Helmut Maier
Affiliation:
Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081Ulm, Germany Email: helmut.maier@uni-ulm.de Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland
Michael Th. Rassias
Affiliation:
Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Institutskiy per, d. 9, RussiaInstitute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ 08540, USA Email: michail.rassias@math.uzh.ch
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Abstract

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One of the approaches to the Riemann Hypothesis is the Nyman–Beurling criterion. Cotangent sums play a significant role in this criterion. Here we investigate the values of these cotangent sums for various shifts of the argument.

Keywords

cotangent sumequidistributionmomentasymptoticsEstermann zeta functionRiemann zeta functionRiemann Hypothesisexponential sums in finite fields

MSC classification

Primary: 33B10: Exponential and trigonometric functions
Secondary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 11M06: $zeta (s)$ and $L(s, chi)$ 28A25: Integration with respect to measures and other set functions 60E10: Characteristic functions; other transforms
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 522 - 535
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2019

Footnotes

M. Th. Rassias was co-funded by the John S. Latsis Public Benefit Foundation and the University of Zürich.

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