Vol. 7, No. 1, 2013

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Period functions and cotangent sums

Sandro Bettin and Brian Conrey

Vol. 7 (2013), No. 1, 215–242
Abstract

We investigate the period function of n=1σa(n)e(nz), showing it can be analytically continued to |argz| < 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