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

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.

6 pages (two minor typos have been corrected)