Ramanujan, Landau et Casimir, les séries divergentes: l’approche d’un physicien
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. 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.
Related
BibTeX
@article{montambaux2026ramanujan,
title = {{Ramanujan, Landau et Casimir, les séries divergentes: l’approche d’un physicien}},
author = {Gilles Montambaux},
journal = {Comptes Rendus. Physique},
year = {2026},
month = nov,
doi = {10.5802/crphys.256},
eprint = {2506.08664},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05853598v1},
note = {CC BY},
}
RIS
TY - JOUR
TI - Ramanujan, Landau et Casimir, les séries divergentes: l’approche d’un physicien
AU - Gilles Montambaux
PY - 2026
DA - 2026/11/05
JO - Comptes Rendus. Physique
DO - 10.5802/crphys.256
UR - https://hal.science/hal-05853598v1
AB - 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.
SN - 1631-0705
SN - 1878-1535
ER -