  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.
