Ramanujan, Landau and Casimir, divergent series: a physicist 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. 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. 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.