From a cotangent sum to a generalized totient function

Abstract

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.