By Klaus Kirsten, Floyd L. Williams
ISBN-10: 0521199301
ISBN-13: 9780521199308
This booklet offers an advent to (1) a variety of zeta capabilities (for instance, Riemann, Hurwitz, Barnes, Epstein, Selberg, and Ruelle), together with graph zeta services; (2) modular types (Eisenstein sequence, Hecke and Dirichlet L-functions, Ramanujan's tau functionality, and cusp forms); and (3) vertex operator algebras (correlation capabilities, quasimodular types, modular invariance, rationality, and a few present study subject matters together with greater genus conformal box theory). a number of concrete functions of the fabric to physics are awarded. those comprise 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 concept.
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Extra resources for A Window Into Zeta and Modular Physics
Sample text
P/. Proceeding inductively, we consider a set fp1 ; p2 ; : : : ; pl ; plC1 g of l C1 distinct primes in P . 3). n/=ns0 converges absolutely. p/. 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).
N/ is the number of positive divisors of n. n/ (remarkably) are all integers. 23, for k D 4; 6. 45), to three-dimensional gravity with a negative cosmological constant will be given in my Speaker’s Lecture; see especially equation (5-8) on page 343 and the subsequent discussion. z/, one cannot take k D 2 for convergence reasons. z/. n/ D 0
Z/ is a modular form of weight k1 C k2 . z/ for 2 , z 2 C . z/ was initially constructed by R. Dedekind in 1877, and by F. Klein in 1878. 41) 42 FLOYD L. z/ 2 S12 . 23, for k D 4; 6. There are no nonzero cusp forms of weight < 12. n/j < C n6 for n 1. However P. n/ is the number of positive divisors of n. n/ (remarkably) are all integers. 23, for k D 4; 6. 45), to three-dimensional gravity with a negative cosmological constant will be given in my Speaker’s Lecture; see especially equation (5-8) on page 343 and the subsequent discussion.
A Window Into Zeta and Modular Physics by Klaus Kirsten, Floyd L. Williams
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