Period functions and cotangent sums

We investigate the period function of ∑n=1∞σa(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.