Negative Sums of Positive Numbers, About Grandi’s Series and a Certain Recurrence of Prime Numbers

Abstract

Already in the 18th century, Euler was known to the following account: ∑n = −1/12 (Ramanujan summation). The above infinite sum of natural numbers applies to certain physical issues, including quantum field theory and string theory (e.g. Casimir effect). From a mathematical point of view, it is associated with the so-called Riemann's zeta function. This article derives the above relationship without referring to the Riemann zeta function and the Ramanujan summation method.