Floyd Williams, professor emeritus, recently published Some musings on Theta, Eta, and Zeta with Springer. The connecting theme is how modular forms and the famous functions in the title appear in theoretical physics, including black holes, exact solutions of Einstein's equations, thermodynamics in curved backgrounds, and more.
Highlights include
- A precise asymptotic formula for the Fourier coefficients of the partition function of a holomorphic conformal field theory with central charge 24k. This formula, which improves a result of Edward Witten, provides for logarithmic corrections to Bekenstein-Cardy-Hawking black hole entropy.
- Connections between black hole and cold plasma physics, including the black holes with a naked singularity.
- An oscillating kink-anti-kink solution of the sine-Gordon equation.
- A canonical zeta function and theta function are assigned to a noncompact Riemannian symmetric space X of arbitrary rank, and when X has complex type,
- It is shown that the short- time asymptotic expansion of theta is actually "exact," and all of the corresponding Minakshisundaram-Pleijel coefficients are explicitly computed.
- The 1-loop effective potential of X is explicitly computed in terms of the special values of zeta and its derivative at 0.
This book is something of a continuation of A window into zeta and modular physics, his prior book.
See https://link.springer.com/book/10.1007/978-981-99-5336-3 for more information.