Period functions and cotangent sums

Abstract

We investigate the period function of ${\sum }_{n=1}^{\infty }{\sigma }_a\left(n\right)e\left(nz\right)$, 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.