El valor de la función zeta de Riemann en los números pares positivos

Abstract

This work focuses on the explicit calculation of the series∑∞ 𝑛�=1 1 𝑛� 2𝑘� , 𝑘� ≥ 1, within a historical framework. It begins with a historical and biographical context, highlighting the Basel Problem’s emergence and Euler’s early solutions. The study considers Euler’s techniques, including methods involving infinitesimal and infinite quantities. It also explores modern approaches using residue theorems and telescoping series, aiming to provide rigor to Euler’s historical work.