48 research outputs found

    From a cotangent sum to a generalized totient function

    Full text link
    In this paper we investigate a certain category of cotangent sums and more specifically the sum βˆ‘m=1bβˆ’1cot⁑(Ο€mb)sin⁑3(2Ο€mab)β€…\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 Ο•(n,A,B)\phi(n,A,B), where Ο•(n,A,B):=βˆ‘A≀k≀B(n,k)=11β€….\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
    corecore