Modular properties of Eisenstein series and statistical physics

TL;DR

This paper explores the modular properties of Eisenstein series, connecting them to statistical physics, elliptic functions, and historical developments in number theory, providing new derivations and interpretations of related formulas.

Contribution

It offers a new perspective on Eisenstein series' modular properties, linking them to statistical physics and revisiting classical formulas with modern techniques.

Findings

Extended temperature inversion properties of internal energy on spheres.

Re-derivation of period functions of Eisenstein series using contour integration.

Reinterpretation of the Epstein--Kober formula and historical Dirichlet series work.

Abstract

The temperature inversion properties of the internal energy, E, on odd spheres, and its derivatives, together with their expression in elliptic terms, as expounded in previous papers, are extended to the integrals of E, thence making contact with the theory of modular forms with rational period functions. I point out that the period functions of (holomorphic) Eisenstein series computed by Zagier were already available since the time of Ramanujan and I give a rederivation by contour integration. Removing both the Planck and Casimir terms gives a fully subtracted form of the series which allows a more elegant and compact treatment. I expound the relation to Eichler cohomology cocycles and also rewrite the theory in a distributional, Green function way. Some historical and technical developments of the Selberg--Chowla formula are presented, and it is suggested that this be renamed the Epstein--Kober formula. On another point of historical justice, the work of Koshliakov on Dirichlet series is reprised. A representation of a `massive' generalised Dirchlet series due to Berndt is also reproved, applied to the Epstein series and to a derivation of the standard statistical mode sum, interpreted as a Kronecker limit formula.