By Klaus Kirsten, Floyd L. Williams
This booklet presents an creation to (1) a number of zeta capabilities (for instance, Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle), together with graph zeta services; (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 kinds, modular invariance, rationality, and a few present study subject matters together with better 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 capabilities, Casimir power calculations, atomic Schrödinger operators, Bose-Einstein condensation, warmth kernel asymptotics, random matrices, quantum chaos, elliptic and theta functionality strategies of Einstein's equations, a soliton-black gap connection in two-dimensional gravity, and conformal box thought.
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Extra info for A Window Into Zeta and Modular Physics
15) P ROOF. 2) once the obvious change of variables v D at is executed. 2n/ t dt D e at 1 a2n 0 for n 2 ޚ, n 1. 2n 1/! m 1/! 15). 15) we shall compute Planck’s radiation density integral. But first we provide some background. On 14 December 1900, a paper written by Max Karl Ernst Ludwig Planck and entitled “On the theory of the energy distribution law of the normal spectrum” was presented to the German Physical Society. That date is considered to be the birthday of quantum mechanics, as that paper set forth for the first time the hypothesis that the energy of emitted radiation is quantized.
Def P ROOF. t CibCm/k on Œ0; 1, for m 2 ޚ. t / converges uniformly on Œ0; 1. 18). 20) 36 FLOYD L. t 2 C 1/k=2 < 1 for k > 1. Since b > 0 is arbitrary, we let b ! 1. k/ D 0. Also for n D 1; 2; 3; 4; : : : and k > 1, it is known that Z k e 2 inx 2 nb . k 1/! x C ib/ see the Remark below. 19). ˜ R EMARK . 22) where ; b > 0, k > 1. x C ib/k D e 2b . k 1/! for ; b > 0 and k > 1 an integer. 21). 11) for the lastPequality) on C . 4), 1 k 1=m C bution to the sum is m2 ޚf0g 1=mk D 1 mD1 1=.
K 1/! 24. f; k/nk 1 for n 1. f; k/nk=2 for n 1. z/ on a fundamental domain F C for the action of on C (given by restriction of the action of G in equation 38 FLOYD L. z/ on C is readily obtained. 14). 27) on C . z/. 23). 15 of , for example, we should add further remarks regarding F . By definition, a fundamental domain for the action of on C is an open set F C such that (F1) no two distinct points of F lie in the same -orbit: if z1 ; z2 2 F with z1 ¤ z2 , then there is no 2 such that z1 D z2 .
A Window Into Zeta and Modular Physics by Klaus Kirsten, Floyd L. Williams