Open Access
2013 Period functions and cotangent sums
Sandro Bettin, John Conrey
Algebra Number Theory 7(1): 215-242 (2013). DOI: 10.2140/ant.2013.7.215


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.


Download Citation

Sandro Bettin. John Conrey. "Period functions and cotangent sums." Algebra Number Theory 7 (1) 215 - 242, 2013.


Received: 1 December 2011; Revised: 15 January 2012; Accepted: 20 February 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1291.11111
MathSciNet: MR3037895
Digital Object Identifier: 10.2140/ant.2013.7.215

Primary: 11M06
Secondary: 11L99 , 11M41

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

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 1 • 2013
Back to Top