Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187-8577, Japan ; Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada K7L 3N6
Abstract:
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 having positive imaginary part. When , these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).