An asymptotic property on a reciprocity law for the Bettin–Conrey cotangent sum

In 2013, Bettin and Conrey introduced a cotangent sum [Formula: see text], which can be regarded as a variant of the Dedekind sum. They discovered that the cotangent sum satisfies a kind of reciprocity law. Roughly speaking, the reciprocity law for [Formula: see text] means that there is a relation between [Formula: see text] and [Formula: see text] modulo holomorphic functions. Furthermore they investigated Taylor coefficients [Formula: see text] of the implicit holomorphic function, which appears in the reciprocity law for [Formula: see text], at [Formula: see text]. As a result, they obtained an asymptotic formula for [Formula: see text] as [Formula: see text]. In this paper we improve it to an asymptotic series expansion. This resolves a conjecture by Zagier. A new ingredient of this paper is to use the confluent hypergeometric function of the second kind.