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

2021/07/15 by Fernando Chamizo, Chamizo, Fernando
Mathematics · #11F67 #11Y60 (Primary) 11F27 (Secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2107.07245

openalex publication_date 2021/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We evaluate the classic sum ∑_n∈ℤ e^-πn2. The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuations. Even the Γ 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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