Elliptic Apostol Sums and Their Reciprocity Laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter τ having positive imaginary part. When τ → i∞, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable τ . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).