Asymptotic formulas for Vasyunin cotangent sums

Abstract

Abstract We study the Vasyunin-type cotangent sum c 0 ( h / k ) = − ∑ m = 1 k − 1 ( m / k ) cot ( π h m / k ) , where h and k are positive coprime integers. This sum is related to Estermann zeta function. By applying the Euler-Maclaurin summation formula to a suitable function, we improve a previous large-k asymptotic approximation of c0(h/k). We also provide a procedure to compute an arbitrary number of terms of the approximation.