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

Fernando Chamizo

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arXiv · 2107.07245

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

Fernando Chamizo

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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.

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Fernando Chamizo. 2021-07-15. A simple evaluation of a theta value and the Kronecker limit formula. https://arxiv.org/abs/2107.07245

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