A simple evaluation of a theta value and the Kronecker limit formula

TL;DR

This paper presents a straightforward evaluation of a classical sum related to the theta function and offers a simple proof of the Kronecker limit formula without relying on advanced concepts like modular forms or elliptic functions.

Contribution

It introduces a novel, elementary approach to evaluate the theta sum and proves the Kronecker limit formula using basic complex analysis techniques.

Findings

Explicit evaluation of the theta sum without modular form knowledge

A simplified proof of the Kronecker limit formula

Demonstration of complex integral methods in classical analysis

Abstract

We evaluate the classic sum $\sum_{n\in\mathbb{Z}} e^{-\pi n^2}$. The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuations. Even the $\Gamma$ function, in terms of which the result is expressed, only appears as a complex function in the computation of a real integral by the residue theorem. Another contribution of this note is to provide a very simple proof of the Kronecker limit formula.