On the Vasyunin Cotangent sums related to Riemann Hypothesis

Abstract: In this work, we are interested by Vasyunin cotangent-sum V (p/q) encountered in computation of the inner product arising in the Baez-Duarte-Balazard criterion for Riemann hypothesis. By hint of generating functions theory and introduction of double Euclidean algorithm, we give series expansions of V (p/q) and the symmetric sum S (p, q) = V (p/q)+V (q/p) . These calculus permit to deduce another reformulation of Vasyunin formula. This study is a complement of the recent work of M. Goubi concerning special case V (1/q).