Zeta Regularization Techniques with Applications

Part 1 The Riemann Zeta function: Riemann, Hurwitz, Epstein, Selberg and related zeta functions analytic continuation - practical uses for series summation asymptotic expansion of zeta. Part 2 Zeta-function regularization of sums over known spectrum: the zeta-function regularization theorem multiple zeta-functions with arbitrary exponents. Part 3 Zeta-function regularization when the spectrum is not known: zeta-function vs heat-kernel regularization small-t asymptotic expansion of the heat-kernel. Part 4 The Casimir effect in flat space-time with compact spatial part: simply connected compact manifold with constant curvature the Selberg trace formula for compact hyperbolic manifolds. Part 5 Finite temperature effects for theories defined on compact hyperbolic manifolds: basic formalism for the finite-temperature effective potential the finite-temperature thermodynamic potential for manifolds with a compact spatial part. Part 6 Properties of the chemical potential in higher-dimensional manifolds: the flat-manifold case the constant non-zero curvature case. Part 7 Strings at non-zero temperature and 2d gravity: free energy for the Bosonic string vacuum energy for Torus compactified strings. Part 8 Membranes at non-zero temperatures: supermembrane free energy free energy for the compactified supermembranes and modular invariance and others.