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 Schrödinger 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.

### Contents

Part I. Introductory Lectures: 1. Lectures on zeta functions, L-functions and modular forms with some physical applications Floyd L. Williams; 2. Basic zeta functions and some applications in physics Klaus Kirsten; 3. Zeta functions and chaos Audrey Terras; 4. Vertex operators and modular forms Geoffrey Mason and Michael Tuite; Part II. Research Lectures: 5. Applications of elliptic and theta functions to Friedmann-Robertson-Lemaître-Walker cosmology with cosmological constant Jennie D'Ambroise; 6. Integrable systems and 2D gravitation: how a soliton illuminates a black hole Shabnam Beheshti; 7. Functional determinants in higher dimensions using contour integrals Klaus Kirsten; 8. The role of the Patterson-Selberg zeta function of a hyperbolic cylinder in three-dimensional gravity with a negative cosmological constant Floyd L. Williams.