A simple evaluation of a theta value, the Kronecker limit formula and a formula of Ramanujan
Autor (es)
Chamizo Lorente, FernandoEntidad
UAM. Departamento de MatemáticasEditor
SpringerFecha de edición
2022-06-09Cita
The Ramanujan Journal (2022): 1-8ISSN
1382-4090 (print); 1572-9303 (online)Financiado por
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NatureVersión del editor
https://doi.org/10.1007/s11139-022-00586-0Materias
Theta function; Kronecker limit formula; Gamma function; MatemáticasDerechos
© The Author(s) 2022Resumen
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
continuation. Even the 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. Finally, employing the evaluation of the sum and some other
ideas, we also obtain an undemanding proof of one of the most emblematic formulas
of Ramanujan
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