From a cotangent sum to a generalized totient function

In this paper we investigate a certain category of cotangent sums and more specifically the sum $$\sum_{m=1}^{b-1}\cot\left(\frac{\pi m}{b}\right)\sin^{3}\left(2\pi m\frac{a}{b}\right)\:$$ and associate the distribution of its values to a generalized totient function $\phi(n,A,B)$, where $$\phi(n,A,B):=\sum_{\substack{A\leq k \leq B \\ (n,k)=1}}1\:.$$ One of the methods used consists in the exploitation of relations between trigonometric sums and the fractional part of a real number

Period functions and cotangent sums

We investigate the period function of \sum_{n=1}^\infty\sigma_a(n)\e{nz}, showing it can be analytically continued to $|\arg z|<\pi$ 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. In particular, we find a reciprocity formula for the Vasyunin sum.Comment: 32 pages, 5 figures, revised version. To appear in Algebra & Number Theor