The Dedekind symbol associated with the Eisenstein series of weight two

Abstract

Dedekind symbols generalize the classical Dedekind sums (symbols). These symbols are determined uniquely, up to additive constants, by their reciprocity laws. For k ≧ 2, there is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws of degree 2k − 2 and the space of modular forms of weight 2k for the full modular group \(PSL_2 (\mathbb{Z}).\) However, this is not the case when k = 1 as there is no modular form of weight two; nevertheless, there exists a unique (up to a scalar multiple) quasi-modular form (Eisenstein series) of weight two. The purpose of this note is to define the Dedekind symbol associated with this quasi-modular form, and to prove its reciprocity law. Furthermore we show that the odd part of this Dedekind symbol is nothing but a scalar multiple of the classical Dedekind sum. This gives yet another proof of the reciprocity law for the classical Dedekind sum in terms of the quasi-modular form.