# From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

## DOI:

https://doi.org/10.31349/RevMexFisE.18.020203## Keywords:

Ramanujan summation, Casimir effect, renormalization, zeta-function## Abstract

A century ago Srinivasa Ramanujan --- the great self-taughtIndian genius of mathematics --- died, shortly after returning

from Cambridge, UK, where he had collaborated with Godfrey Hardy.

Ramanujan contributed numerous outstanding results to different

branches of mathematics, like analysis and number theory,

with a focus on special functions and

series. Here we refer to apparently weird values

which he assigned to two simple divergent series, $\sum_{n \geq 1} n$

and $\sum_{n \geq 1} n^{3}$. These values are sensible, however,

as analytic continuations, which correspond to Riemann's

$\zeta$-function. Moreover, they have applications in physics:

we discuss the vacuum energy of the photon field, from which one

can derive the Casimir force, which has

been experimentally measured.

We discuss its interpretation, which remains controversial.

This is a simple way to illustrate the concept of renormalization,

which is vital in quantum field theory.

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*Rev. Mex. Fis. E*, vol. 18, no. 2 Jul-Dec, pp. 020203 1–15, Jul. 2021.

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Copyright (c) 2021 Wolfgang Bietenholz

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