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

Gilles Montambaux

doi:10.5802/crphys.256

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arXiv · 2506.08664

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

Gilles Montambaux

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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.

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Gilles Montambaux. 2025-06-10. Ramanujan, Landau and Casimir, divergent series: a physicist point of view. https://doi.org/10.5802/crphys.256

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