By Klaus Kirsten, Floyd L. Williams

ISBN-10: 0521199301

ISBN-13: 9780521199308

This ebook presents an advent to (1) numerous zeta capabilities (for instance, Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle), together with graph zeta features; (2) modular types (Eisenstein sequence, Hecke and Dirichlet L-functions, Ramanujan's tau functionality, and cusp forms); and (3) vertex operator algebras (correlation features, quasimodular kinds, modular invariance, rationality, and a few present examine subject matters together with better genus conformal box theory). a number of concrete purposes of the fabric to physics are awarded. those contain Kaluza-Klein additional dimensional gravity, Bosonic string calculations, an summary Cardy formulation for black gap entropy, Patterson-Selberg zeta functionality expression of one-loop quantum box and gravity partition capabilities, Casimir strength calculations, atomic Schrödinger operators, Bose-Einstein condensation, warmth kernel asymptotics, random matrices, quantum chaos, elliptic and theta functionality options of Einstein's equations, a soliton-black gap connection in two-dimensional gravity, and conformal box idea.

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D ރf0g. For our purpose, however, there is an LECTURES ON ZETA FUNCTIONS, L-FUNCTIONS AND MODULAR FORMS 47 equivalent way of thinking about characters modulo m. In fact, given , define ޚW ރ ! 1) 0 otherwise, for n 2 ޚ. n2 ; m/ D 1. Conversely, suppose 0 W ރ ! ޚis a function that satisfies the three conditions (D1), (D2), and (D3). Define W Um ! n; m/ D 1. The character is well-defined by (D2), and W Um ! ރby (D1). b/ for a; b 2 Um , so we see that is a character modulo m.

12). ˜ 26 FLOYD L. WILLIAMS Now let pl be the l-th positive prime, so p1 D 2, p2 D 3, p3 D 5, p4 D 7, . . n/ ns0 ; since no pj divides 1. 15) Namely, for n 2 choose q 2 P such that q j n. If no pj divides n, 1 j l, then q ¤ p1 ; : : : ; pl (since q j n); hence q plC1 (since pl is the l-th prime) and so q l C 1. But this is impossible since n l (by hypothesis) and q n (since q j n). 15). n/ ˇ P ˇ ˇ ˇ ns0 ˇ : nD1 But this difference tends to 0 as l ! 1. 1/. 16) We have therefore finally reached the main theorem.

D Sk . 28). z/ is not a cusp form. If f 2 Mk . 24 this series converges absolutely for Re s > k, and for Re s > 1C k2 if f 2 Sk . /. sI f / is holomorphic in s (by an argument similar to that for the Riemann zeta function), as we have asserted in Lecture 3. 32. Suppose the Fourier coefficients an of f 2 Mk . 33) 40 FLOYD L. WILLIAMS for n1 ; n2 1. 34) p2P for Re s > k. If f 2 Sk . 34) holds for Re s > 1 C k=2. Here, we only need to note that a1 D 1. 33) an a1 D an1 =12 D an and so a1 D 1. 33).

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