A Window into Zeta and Modular Physics

This book provides an introduction to (1) various zeta functions (for example Riemann Hurwitz Barnes Epstein Selberg and Ruelle) including graph zeta functions; (2) modular forms (Eisenstein series Hecke and Dirichlet L-functions Ramanujan''s tau function and cusp forms); and (3) vertex operator algebras (correlation functions quasimodular forms modular invariance rationality and some current research topics including higher genus conformal field theory). Various concrete applications of the material to physics are presented. These include Kaluza-Klein extra dimensional gravity Bosonic string calculations an abstract Cardy formula for black hole entropy Patterson-Selberg zeta function expression of one-loop quantum field and gravity partition functions Casimir energy calculations atomic Schrdinger operators Bose-Einstein condensation heat kernel asymptotics random matrices quantum chaos elliptic and theta function solutions of Einstein''s equations a soliton-black hole connection in two-dimensional gravity and conformal field theory.