Ramanujan, Landau and Casimir, divergent series: a physicist point of view

TL;DR

This paper explores the physical significance of the divergent series sum of positive integers equaling -1/12, demonstrating its relevance in condensed matter and quantum electrodynamics through a physicist's qualitative approach.

Contribution

It introduces a systematic physicist-oriented method to interpret and extract the -1/12 sum from divergent series in physical contexts.

Findings

The -1/12 sum appears in Landau diamagnetism.

The -1/12 sum is relevant in the Casimir effect.

A qualitative approach to divergent series is proposed.

Abstract

It is a popular paradoxical exercise to show that the infinite sum of positive integer numbers is equal to -1/12, sometimes called the Ramanujan sum. Here we propose a qualitative approach, much like that of a physicist, to show how the value -1/12 can make sense and, in fact, appears in certain physical quantities where this type of summation is involved. At the light of two physical examples, taken respectively from condensed matter -- the Landau diamagnetism -- and quantum electrodynamics -- the Casimir effect -- that illustrate this strange sum, we present a systematic way to extract this Ramanujan term from the infinity.