Explicit and asymptotic formulae for Vasyunin-cotangent sums

For coprime numbers p and q, we consider the Vasyunin–cotangent sum (0.1) V (q, p) = p−1 ∑ k=1 {kq p } cot (πk p ) . First, we prove explicit formula for the symmetric sum V (p, q)+V (q, p) which is a new reciprocity law for the sums (0.1). This formula can be seen as a complement to the Bettin–Conrey result [13, Theorem 1]. Second, we establish asymptotic formula for V (p, q). Finally, by use of continued fraction theory, we give formula for V (p, q) in terms of continued fraction of p q .