Period functions and cotangent sums

We investigate the period function of $\sum_{n=1}^\inftyσ_a(n)\e{nz}$, showing it can be analytically continued to $|\arg z|<π$ 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. In particular, we find a reciprocity formula for the Vasyunin sum.

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