Riemann zeta function

Abstract: This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.

Abstract: 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.

Abstract: We derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices. This ratio distribution was recently introduced to study spectral properties of many-body problems, as, contrary to the standard level spacing distributions, it does not depend on the local density of states. Our Wigner-like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit. Quantitative improvements are found through a polynomial expansion. Examples from a quantum many-body lattice model and from zeros of the Riemann zeta function are presented.