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 $\varphi (n,A,B)$, where $$\varphi (n,A,B):=\sum _{\begin{array}{c}A\le k\le B\\ (n,k)=1\end{array}}1.$$ One of the methods used consists in the exploitation of relations between trigonometric sums and the fractional part of a real number.