#### 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 ${\sum }_{n=1}^{\infty }{\sigma }_{a}\left(n\right)\phantom{\rule{0.3em}{0ex}}e\left(nz\right)$, showing it can be analytically continued to $|argz|<\pi$ 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
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