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

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.

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
Received: 1 December 2011
Revised: 15 January 2012
Accepted: 20 February 2012
Published: 28 March 2013
Sandro Bettin
School of Mathematics
University of Bristol
Howard House
Queens Avenue
Bristol BS82NF
United Kingdom
Brian Conrey
American Institute of Mathematics
360 Portage Avenue
Palo Alto, CA 94306
United States