By Klaus Kirsten, Floyd L. Williams

This ebook presents an advent to (1) quite a few zeta services (for instance, Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle), together with graph zeta features; (2) modular varieties (Eisenstein sequence, Hecke and Dirichlet L-functions, Ramanujan's tau functionality, and cusp forms); and (3) vertex operator algebras (correlation features, quasimodular varieties, modular invariance, rationality, and a few present study issues together with greater genus conformal box theory). a number of concrete purposes of the fabric to physics are offered. 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 services, Casimir power calculations, atomic Schrödinger operators, Bose-Einstein condensation, warmth kernel asymptotics, random matrices, quantum chaos, elliptic and theta functionality suggestions of Einstein's equations, a soliton-black gap connection in two-dimensional gravity, and conformal box thought.

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Z/ is not a modular form of weight 2. 46). z/. 57). w/ D 1 C w on D. w/ D def 1 n =n. z/ D e 2 iz , z 2 C , consider the n-th partial nD1 . z/ D nkD1 g. z/k 2 D for k > 0. z/ D kD1 g. 58) P1 P k n converges. 54), This allows P1 for example. z/ kD1 nD1 nD1 kD1 Q1 the finiteness of these series. z/ is finite. 1 Qn Qn k/ k g. 58). z/ D 1 X 0 g . z/ /. n/e 2 inz ; 46 FLOYD L. 52). 46). z/; z where we take arg z 2 . ; /. z/=4 i C . 57). z/. z/. k/. Lecture 5. 19) has an application to Dirichlet L-functions, which we now consider.

8). This concludes the proof. aCbx/ Cb y D 2 D 1 > 12 ; which Now suppose b D 0, but a ¤ 0. 0; 0/g. m2 Cn2 /K1 D ˛=2 K1 jm C nij2 . m;n/2 ޚ2 1=jm C nij˛ converges for ˛ > 2. 4). z/ on C is established. n; m/ is a bijection of ޚ2 . z/ satisfies the conditions (M1)0 and (M1)00 . z/, for k even 4, are modular forms of weight k, we must check condition (M2). z/ are actually computed explicitly. 11) 34 FLOYD L. WILLIAMS on C for k 2 ޚ, k 2. 12) for z 2 SA;ı . 14) for n 2 ޚ, b > 0.

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. 0 / ޚW ރ ! 1) coincides with 0 . n2 / for all n1 ; n2 2 ޚ. n1 n2 / are zero. n2 / by (D3). 1/ N D 1. 1), which in turn equals 1. n/j 1 for all n 2 ޚ. The proof of (D5) makes use of a little theorem in group theory which says that if G is a finite group with jGj elements, then ajGj D 1 for every a 2 G.