Gilles Montambaux : Ramanujan, Landau and Casimir, divergent series: A physicist’s approach

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. This result is actually well-defined in a proper mathematical sense.
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.
In 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 the Ramanujan term from the infinity.
In both examples, the ”infinite” appears to be a vacuum energy and the Ramanujan sum is revealed by a response function to an external parameter.