Explicit and asymptotic formulae for Vasyunin-Cotangent Sums

Abstract

For coprime numbers p and q, we consider the Vasyunin-cotangent sum First, we prove explicit formula for the symmetric sum V (p, q)+V (q, p) which is a new reciprocity law for the sums above. This formula can be seen as a complement to the Bettin-Conrey result [13, Theorem 1]. Second, we establish an asymptotic formula for V (p, q). Finally, by use of continued fraction theory, we give a formula for V (p, q) in terms of continued fraction of p/q.