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

Published online by Cambridge University Press:  15 October 2019

Helmut Maier
Affiliation:
Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081 Ulm, 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, Russia Institute for Advanced Study, Program in Interdisciplinary Studies, 1 Einstein Dr, Princeton, NJ 08540, USA Email: michail.rassias@math.uzh.ch
Corresponding
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Abstract

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.

Type
Article
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.

References

Bagchi, B., On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis . Proc. Indian Acad. Sci. Math. 116(2006), 137146. https://doi.org/10.1007/BF02829783 CrossRefGoogle Scholar
Bettin, S., A generalization of Rademacher’s reciprocity law . Acta Arith. 159(2013), 363374. https://doi.org/10.4064/aa159-4-5 CrossRefGoogle Scholar
Bettin, S., On the distribution of a cotangent sum . IMRN 2015 no. 21, https://doi.org/10.1093/imrn/rnv036 Google Scholar
Bettin, S. and Conrey, B., Period functions and cotangent sums . Algebra Number Theory 7(2013), 215242. https://doi.org/10.2140/ant.2013.7.215 CrossRefGoogle Scholar
de la Bretèche, R. and Tenenbaum, G., Séries trigonométriques à coefficients arithmétiques . J. Anal. Math. 92(2004), 179. https://doi.org/10.1007/BF02787756 CrossRefGoogle Scholar
Fouvry, E. and Michel, Ph., Sur certaines sommes d’exponentielles sur les nombres premiers . Ann. Sci. Écope Norm. Sup. (4) 31(1998), 93130. https://doi.org/10.1016/S0012-9593(98)80019-0 CrossRefGoogle Scholar
Maier, H. and Rassias, M. Th., The order of magnitude for moments for certain cotangent sums . J. Math. Anal. Appl. 429(2015), 576590. https://doi.org/10.1016/j.jmaa.2015.04.036 Google Scholar
Maier, H. and Rassias, M. Th., Generalizations of a cotangent sum associated to the Estermann zeta function . Commun. Contemp. Math. 18(2016), no. 1, 1550078. https://doi.org/10.1142/S0219199715500789 CrossRefGoogle Scholar
Maier, H. and Rassias, M. Th., The rate of growth of moments of certain cotangent sums . Aequationes Math. 90(2016), 581595. https://doi.org/10.1007/s00010-015-0361-3 CrossRefGoogle Scholar
Maier, H. and Rassias, M. Th., Asymptotics for moments of certain cotangent sums . Houston J. Math. 43(2017), 207222.Google Scholar
Maier, H. and Rassias, M. Th., The maximum of cotangent sums related to Estermann’s zeta function in rational numbers in short intervals . Appl. Anal. Discrete Math. 11(2017), 166176. https://doi.org/10.2298/AADM1701166M CrossRefGoogle Scholar
Maier, H. and Rassias, M. Th., Distribution of a cotangent sum related to the Nyman–Beurling criterion for the Riemann Hypothesis . Appl. Math. Comput. 363(2019), 124589. https://doi.org/10.1016/j.amc.2019.124589 Google Scholar
Rassias, M. Th., Analytic investigation of cotangent sums related to the Riemann zeta function, Doctoral Dissertation, ETH-Zürich, Switzerland, 2014.Google Scholar
Rassias, M. Th., On a cotangent sum related to zeros of the Estermann zeta function . Appl. Math. Comput. 240(2014), 161167. https://doi.org/10.1016/j.amc.2014.04.086 Google Scholar
Weil, A., On some exponential sums . Proc. Natl. Acad. Sci. USA 34(1948), 204207. https://doi.org/10.1073/pnas.34.5.204 CrossRefGoogle ScholarPubMed
Weil, A., Sur les courbes algébriques et les variétés qui s’en déduisent . Actualités Sci. Ind. 1041. Publ. Inst. Math. Univ. Strasbourg 7(1945). Paris, Hermann 1948.Google Scholar
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