## A Window Into Zeta and Modular PhysicsThis 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. |

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### Contents

Basic zeta functions and some applications in physics | 101 |

Zeta functions and chaos | 145 |

Analytic continuation and functional equation of | 146 |

Vertex operators and modular forms | 183 |

Research Lectures | 279 |

How a soliton illuminates a black hole | 295 |

Functional determinants in higher dimensions using contour integrals | 307 |

since j1nsj D 1nRes 1n1Cı on Sı with | 319 |

The role of the PattersonSelberg zeta function of a hyperbolic cylinder | 329 |

### Common terms and phrases

ˇ ˇ ˇ 1-point functions adjacency matrix asymptotic black hole Bose–Einstein condensation bosonic boundary conditions C2-coﬁnite Casimir energy Casimir force central charge compute conformal consider contour converges absolutely correlation functions deﬁned deﬁnition denote determinant formula dimension edge zeta eigenvalues Eisenstein series elliptic functions entropy example EXERCISE ﬁeld equations ﬁeld theory ﬁnd ﬁnite ﬁrst follows form of weight Fourier coefﬁcients Fourier expansion genus-two given graph heat kernel Heisenberg VOA Ihara zeta function inﬁnite integral Jacobi lattice lectures Lemma Lie algebra linear Math Mathematical meromorphic metric modular forms Note number theory obtain partition function Phys physics poles polynomial prime number theorem proof properties Prove radius rational representation result Riemann hypothesis Riemann zeta function satisﬁes scalar Section solution Terras theta V-module vector vector-valued modular form vertex algebra vertex operator algebras Virasoro algebra Virc zero