Seminar

We consider the sum $\sum_{n=1}^\infty \sigma_a(n)e^{2\pi i nz}$, showing that its period function can be analytically continued in z and has a very fast converging Taylor series in Re(z)>0. We then use these results to deduce an exact formula for the second moment of the Riemann zeta function. Moreover, we introduce a family of cotangent sums, functions over the rationals that generalize the Dedekind sum, and share with it the property of having an "almost" analytic period function.