Topic
Riemann zeta function
About: Riemann zeta function is a research topic. Over the lifetime, 6905 publications have been published within this topic receiving 109297 citations. The topic is also known as: Euler-Riemann zeta function & zeta function.
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01 Jun 1995
TL;DR: In this article, the saddle-point method was used for arithmetic progressions, and the Euler gamma function and the Riemann zeta function were used to generate arithmetic functions.
Abstract: Elementary methods Some tools from real analysis Prime numbers Arithmetic functions Average orders Sieve methods Extremal orders The method of van der Corput Diophantine approximation Complex analysis methods The Euler gamma function Generating functions: Dirichlet series Summation formulae The Riemann zeta function The prime number theorem and the Riemann hypothesis The Selberg-Delange method Two arithmetic applications Tauberian theorems Primes in arithmetic progressions Probabilistic methods Densities Limiting distributions of arithmetic functions Normal order Distribution of additive functions and mean values of multiplicative functions Friable integers The saddle-point method Integers free of small factors Bibliography Index
1,314 citations
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TL;DR: In this article, a generalized zeta function was proposed to regularize quadratic path integrals on a curved background spacetime, which 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.
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.
1,251 citations
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01 May 1994
TL;DR: In this article, the Riemann Zeta function analytic continuation is used for series summation asymptotic expansion of "zeta", and the Casimir effect in flat space-time with compact spatial part.
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.
1,032 citations
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01 Jan 1992TL;DR: For Re s = σ > 1, the Riemann zeta function ζ(s) is defined by as discussed by the authors, and it follows from the definition that ζ is an analytic function in the halfplane Re s > 1.
Abstract: For Re s = σ > 1, theRiemann zeta function ζ(s)is defined by
$$ \zeta \left( s \right) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^s}}}} . $$
It follows from the definition that ζ(s) is an analytic function in the half-plane Re s > 1.
1,000 citations