2025/06/10 by Montambaux, Gilles
#FOS: Physical sciences #Mesoscale and Nanoscale Physics (cond-mat.mes-hall) #Other Condensed Matter (cond-mat.other)
paper · doi:10.48550/arxiv.2506.08664
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.