We investigate the period function of , showing it can be analytically continued to 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.
Sandro Bettin. John Conrey. "Period functions and cotangent sums." Algebra Number Theory 7 (1) 215 - 242, 2013. https://doi.org/10.2140/ant.2013.7.215