Showing papers on "Riemann zeta function published in 1977"

Zeta function regularization of path integrals in curved spacetime

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.

On the asymptotic distribution of closed geodesics on compact Riemann surfaces

Abstract: The set of lengths of closed geodesics on a compact Riemann surface is related to the Selberg zeta function in a manner which is evocative of the relationship between the rational primes and the Riemann zeta function. In this paper, this connection is developed to derive results about the asymptotic distribution of these lengths. Suppose S is a compact Riemann surface, regarded as a quotient of the upper half-plane H + by a discontinuous group r. We assume that H + is endowed with the metric y 2((dx)2 + (dy)2), and we denote the volume of S by A. As is well known, the closed geodesics on S are in one-to-one correspondence with the conjugacy classes of r. In particular, if y is an element of r other than the identity, it can, by a change of coordinates, be put into the nonnal form z -N.z, with N. > 1. The number Ny, which is evidently the same within a conjugacy class, is called the norm of the element. The quantity log N., is then the length of the closed geodesic corresponding to the conjugacy class of 'y (see, e.g., [4]). Huber [3] has investigated the asymptotic distribution of the closed geodesics, and found that their statistical behavior is heavily influenced by the ."small" eigenvalues of the Laplace operator on S. That is, by the eigenvalues, if any, in (0, 1 ) for the problem Af + Af = 0 on S. Now questions of this sort* can be very easily treated using the standard theory of the Selberg zeta function, and from this point of view, the analysis is very evocative of the relationship between the distribution of rational primes and the Riemann zeta function. In this paper, we will outline such a treatment. We begin with a few definitions. From now on, the symbol y will represent a closed geodesic on S. The symbol y" will mean the geodesic obtained from y by n-fold iteration. A closed geodesic will be called primitive if it is not a positive integral power of any geodesic other than itself. By analogy with the theory of rational primes, we will define A(y) to be log N.,, where y = yo', with yo primitive. Received by the editors April 27, 1976. AMS (MOS) subject classifications (1970). Primary 30A46, 58F20, 30A58. C American Mathematical Society 1977 241 This content downloaded from 207.46.13.169 on Sat, 01 Oct 2016 06:05:45 UTC All use subject to http://about.jstor.org/terms

High Precision Coefficients Related to the Zeta Function.

Abstract: : If one wishes to calculate numerical approximations for the Riemann zeta function in the 'critical strip', the most efficient procedure presently known is to use the Riemann-Siegel Formula. In the present report, highly accurate approximations for the primary coefficients appearing in this formula are given.

Application of Analytic Regularization to the Casimir Forces

Abstract: Analytic regularization for the Casimir Effect for rectangular systems in one-, two- and three-dimensions, as well as for parallel conducting plates, is discussed. We consider the analytic regularization by employing the Riemann zeta function as well as the zeta functions introduced by Epstein. The forces, in this case, come out automatically finite, i. e., no subtractions are needed. We show that the analytic continuation, in the number of imaginary time dimensions, corresponds to introducing generalized zeta functions for the zero point energy. Discutimos a regularizacao analitica para o Efeito Casimir em sistemas retangulares a uma, duas e tres dimensoes, assim como para placas condutoras e paralelas. No processo de re g ularizacao anal itica fazemos uso da funcao zeta de Riemann assim como das funcoes zeta introduzidas por Epstein. As forcas, neste caso, sao automaticamente f ini tas, sem a necessidade de subtracoes. Mostramos que a continuacao anal it ica, no numero de dimensoes temporais imaginarias, corresponde a introducao de funcoes zeta generalizadas para a energia do ponto zero.

Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

Abstract: The following theorem is going to be proved. Letp m be them-th prime and putd m :=p m+1−p m . LetN(σ,T), 1/2≤σ≤1,T≥3. denote the number of zeros ϱ=β+iγ of the Riemann zeta function which fulfill β≥σ and |γ|≤T. Letc≥2 andh≥0 be constants such thatN(σ,T)≪T c(1−σ) (logT) h holds true uniformly in 1/2≤σ≤1. Let e>0 be given. Then there is some constantK>0 such that

Characteristic for reflexive relations

Abstract: Poincare characteristic for reflexive relations (oriented graphs) is defined in terms of homology and is not invariant under transitive closure. Formulas for the Poincare characteristic of products, joins, and bounded products are given. Euler's definition of characteristic extends to certain filtrations of reflexive relations which exist iff there are no oriented loops. Euler characteristic is independent of filtration, agrees with Poincare characteristic, and is unique. One-sided Mobius characteristic is defined as the sum of all values of a one-sided inverse of the zeta function. Such one-sided inverses exist iff there are no local oriented loops (although there may be global oriented loops). One-sided Mobius characteristic need not be Poincare characteristic, but it is when a one-sided local transitivity condition is satisfied. A two-sided local transitivity condition insures the existence of the Mobius function but Mobius inversion fails for non-posets.

The Dedekind Zeta Function and the Class Number Formula

Abstract: We will use the results of chapter 6 to define and establish properties of the Dedekind zeta function of a number ring R. This is a generalization of the familiar Riemann zeta function, which occurs when R = ℤ. Using this function we will determine densities of certain sets of primes and establish a formula for the number of ideal classes in an abelian extension of Q.

The \mathfrak{p}-ADIC Zeta-Function of AN Imaginary Quadratic Field and the Leopoldt Regulator

Abstract: This paper gives a construction of the -adic zeta-function of an imaginary quadratic field which can be used to express the class number with conductor of complex multiplication fields.We obtain an exact formula for the norm of the Leopoldt regulator of such fields; this formula follows from the existence of a -module associated to the regulator.Bibliography: 9 titles.

p-adic interpolation of the Riemann zeta-function

Abstract: This chapter is logically independent of the following chapters, and is presented at this point in the middle of our ascent to Ω as a plateau in the level of abstraction—namely, everything in this chapter still takes place in the fields ℚ, ℚ p , and ℝ.