Hypergeometric formulas for lattice sums and Mahler measures

TL;DR

This paper establishes explicit formulas connecting hypergeometric functions to lattice sums and explores their relation to Mahler measures and elliptic curve L-series, advancing understanding in number theory and special functions.

Contribution

It provides new explicit formulas linking hypergeometric functions with lattice sums and Mahler measures, supporting conjectures on elliptic curves.

Findings

Derived explicit hypergeometric formulas for lattice sums.

Connected Mahler measures to special values of L-series.

Supported Boyd's conjectures on Mahler measures and elliptic curves.

Abstract

We prove a variety of explicit formulas relating special values of generalized hypergeometric functions to lattice sums with four indices of summation. These results are related to Boyd's conjectured identities between Mahler measures and special values of $L$-series of elliptic curves.