Algebra & Number Theory

Period functions and cotangent sums

Sandro Bettin and John Conrey

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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.

Article information

Algebra Number Theory, Volume 7, Number 1 (2013), 215-242.

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

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Zentralblatt MATH identifier

Primary: 11M06: $\zeta (s)$ and $L(s, \chi)$
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11L99: None of the above, but in this section

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


Bettin, Sandro; Conrey, John. Period functions and cotangent sums. Algebra Number Theory 7 (2013), no. 1, 215--242. doi:10.2140/ant.2013.7.215.

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