A Direct Evaluation of the Periods of the Weierstrass Zeta Function

TL;DR

This paper presents a direct method to evaluate the periods of the Weierstrass Zeta function by selecting an optimal summation order, also revealing the quasi-modularity of related Eisenstein series.

Contribution

It introduces a straightforward approach to compute the difference function of the Weierstrass Zeta function and derives the quasi-modularity of weight 2 Eisenstein series from this framework.

Findings

Direct evaluation method for Weierstrass Zeta function periods

Immediate derivation of quasi-modularity of Eisenstein series

Connection between difference function and modular properties

Abstract

We show how to obtain the difference function of the Weierstrass Zeta function very directly, by choosing an appropriate order of summation for the series defining this function. As a byproduct, we show how to obtain the quasi-modularity of the weight 2 Eisenstein series immediately from the fact that it appears in this difference function and the homogeneity properties of the latter.