Period functions and cotangent sums

Bettin, Sandro; Conrey, John

doi:10.2140/ant.2013.7.215

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Period functions and cotangent sums

Sandro Bettin and Brian Conrey

Vol. 7 (2013), No. 1, 215–242

DOI: 10.2140/ant.2013.7.215

Abstract

We investigate the period function of $\sum_{n = 1}^{\infty} σ_{a} (n) e (n z)$ , showing it can be analytically continued to $| arg z | < π$ and studying its Taylor series. We use these results to give a simple proof of the Voronoi formula and to prove an exact formula for the second moments of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions defined over the rationals, that generalize the Dedekind sum and share with it the property of satisfying a reciprocity formula.

Keywords

period functions, moments, mean values, Riemann zeta function, Eisenstein series, Voronoi formula, cotangent sums, Vasyunin sum, Dedekind sum

Mathematical Subject Classification 2010

Primary: 11M06

Secondary: 11M41, 11L99

Milestones

Received: 1 December 2011

Revised: 15 January 2012

Accepted: 20 February 2012

Published: 28 March 2013

Authors

Sandro Bettin
School of Mathematics University of Bristol Howard House Queens Avenue Bristol BS82NF United Kingdom
http://www.maths.bris.ac.uk/~maxsb/
Brian Conrey
American Institute of Mathematics 360 Portage Avenue Palo Alto, CA 94306 United States

Vol. 7, No. 1, 2013