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

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.