Showing papers on "Riemann zeta function published in 2004"
TL;DR: In this article, an analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture, in which the intersection of X and the hypersurface f = 0 is smooth.
Abstract: Let X be a smooth quasiprojective subscheme of Pn of dimension m iÝ 0 over Fq. Then there exist homogeneous polynomials f over Fq for which the intersection of X and the hypersurface f = 0 is smooth. In fact, the set of such f has a positive density, equal to ?AEX(m+1).1, where ?AEX(s) = ZX(q.s) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.
294 citations
79 citations
TL;DR: Using the Weyl quantization, the authors formulated one-dimensional adelic quantum mechanics, which unifies and treats ordinary and $p$-adic quantum mechanics on an equal footing.
Abstract: Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and $p$-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of a simplest vacuum state leads to the well known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested.
62 citations
TL;DR: In this paper, the integer-order derivatives of the Riemann xi function were evaluated and the even order derivatives at s = 0, s = ½, and s = 1 were shown to be positive.
52 citations
13 Jun 2004
TL;DR: In this article, a survey of p-adic cohomology for machine computation of zeta functions of algebraic varieties over finite fields of small characteristic is presented, and some new avenues for further exploration are explored.
Abstract: We survey some recent applications of p-adic cohomology to machine computation of zeta functions of algebraic varieties over finite fields of small characteristic, and suggest some new avenues for further exploration.
46 citations
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TL;DR: In this article, the relation between the zeta function of a Calabi-Yau hypersurface and its mirror is studied, and two types of arithmetic relations are discovered.
Abstract: In this paper, we study the relation between the zeta function of a Calabi-Yau hypersurface and the zeta function of its mirror. Two types of arithmetic relations are discovered. This motivates us to formulate two general arithmetic mirror conjectures for the zeta functions of a mirror pair of Calabi-Yau manifolds.
44 citations
43 citations
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TL;DR: In this paper, the Riemann hypothesis holds for all truncation integrals with truncation parameter T \ge 1, except possibly for two real zeros, which are present if and only if y > 4 \pi e^{-\gamma} = 7.0555+.
Abstract: This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical line Re(s) = 1/2. For the constant term of the Eisenstein series it shows that all zeros are on the critical line for fixed y= Im(z) \ge 1, except possibly for two real zeros, which are present if and only if y > 4 \pi e^{-\gamma} = 7.0555+. It shows the Riemann hypothesis holds for all truncation integrals with truncation parameter T \ge 1. For T=1 this proves the Riemann hypothesis for a zeta function recently introduced by Lin Weng, attached to rank 2 semistable lattices over the rationals.
36 citations
TL;DR: In this article, a Voronoi-style summation formula for the coefficients of a cusp form on GL(3,Z)⧹GL (3,R) is presented.
36 citations
TL;DR: In this paper, the Bruhat-Tits building is used to enumerate the vertices in a group and derive local normal zeta functions for groups of class 2 with small centres.
Abstract: We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sln(Qp). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions.
35 citations
TL;DR: In this paper, the Fourier Transform and the Riemann zeta function were studied in the context of Dirichlet-Dedekind-Hecke-Tate L-functions.
Abstract: We study some of the interactions between the Fourier Transform and the Riemann zeta function (and Dirichlet-Dedekind-Hecke-Tate L-functions).
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TL;DR: In this paper, the authors calculate the total derived functor for the map from the Weil-etale site introduced by Lichtenbaum to the etale site for varieties over finite fields.
Abstract: We calculate the total derived functor for the map from the Weil-etale site introduced by Lichtenbaum to the etale site for varieties over finite fields. In particular, there is a long exact sequence relating Weil-etale cohomology and etale cohomology. In the second half of the paper, we apply this to study the Weil-etale cohomology of the motivic complex for smooth and projective varieties. These groups are expected to be finitely generated, to give an integral model for l-adic cohomology, and to be related to special values of the zeta function. We give necessary and sufficient conditions for this to hold, and examples.
TL;DR: Some evaluation formulas for Tornheim's type of alternating series are given by an elementary and combinatorial calculation of the uniformly convergent series by means of Riemann's zeta values at positive integers.
Abstract: In this paper, we give some evaluation formulas for Tornheim's type of alternating series by an elementary and combinatorial calculation of the uniformly convergent series. Indeed, we list several formulas for them by means of Riemann's zeta values at positive integers.
TL;DR: In this article, a commutative generalization of complex numbers, called bicomplex Riemann numbers, was introduced, and a holomorphic Riemmann zeta function of two complex variables satisfying the complexified Cauchy-Riemann equations was established.
Abstract: In this work we use a commutative generalization of complex numbers, called bicomplex
numbers, to introduce a holomorphic Riemann zeta function of two complex variables
satisfying the complexified Cauchy-Riemann equations Furthermore, we establish a
bicomplex Riemann hypothesis equivalent to the complex Riemann hypothesis of one variable
and we obtain a bicomplex Euler Product
TL;DR: In this article, an analogue of Euler's constant for the Selberg zeta functions of a compact Riemann surface and the Dedekind zeta function of an algebraic number field is studied.
Abstract: The purpose of this paper is to study an analogue of Euler's constant for the Selberg zeta functions of a compact Riemann surface and the Dedekind zeta function of an algebraic number field. Especially, we establish similar expressions of such Euler's constants as de la Vall\'ee-Poussin obtained in 1896 for the Riemann zeta function. We also discuss, so to speak, higher Euler's constants and establish certain formulas concerning the power sums of essential zeroes of these zeta functions similar to Riemann's explicit formula.
TL;DR: In this paper, a modified q-analogue of Riemann zeta function is defined, and q -Bernoulli numbers are interpolated at negative integers in the same way that RiemANN interpolates Bernoulli number at negativeintegers.
Abstract: We consider the modified q -analogue of Riemann zeta function
which is defined by
ζ q ( s ) = ∑ n = 1 ∞ ( q n ( s − 1 ) / [ n ] s ) , 0 q 1 , s ∈ ℂ . In this paper, we give q -Bernoulli numbers
which can be viewed as interpolation of the above q -analogue of
Riemann zeta function at negative integers in the same way that
Riemann zeta function interpolates Bernoulli numbers at negative
integers. Also, we will treat some identities of q -Bernoulli
numbers using nonarchimedean q -integration.
TL;DR: An easily accessible integral representation of the partial sum L"u(x,a)=@?"0"=<"n"= <"x(n+a)^u" of the Hurwitz Zeta function @z(-u,a) is presented, which entails a number of important implications for L"i(x) and @z(n) and for their derivatives.
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TL;DR: In this paper, a quantum algorithm for determining the zeta function of a genus g curve over a finite field F_q, which is polynomial in g and log(q), was presented.
Abstract: We exhibit a quantum algorithm for determining the zeta function of a genus g curve over a finite field F_q, which is polynomial in g and log(q). This amounts to giving an algorithm to produce provably random elements of the class group of a curve, plus a recipe for recovering a Weil polynomial from enough of its cyclic resultants. The latter effectivizes a result of Fried in a restricted setting.
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TL;DR: In this article, a spectral interpretation of the Riemann zeta function was constructed for a nuclear Frechet space whose spectrum is the set of non-trivial zeros of zeta.
Abstract: Based on work of Alain Connes, I have constructed a spectral interpretation for zeros of L-functions. Here we specialise this construction to the Riemann zeta function. We construct an operator on a nuclear Frechet space whose spectrum is the set of non-trivial zeros of zeta. We exhibit the explicit formula for the zeros of the Riemann zeta function as a character formula.
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TL;DR: In this paper, a variational approach was used to find new series representations for well known mathematical constants, such as the Riemann zeta function and the Catalan constant, which are all exponentially convergent and provide useful analytical approximations.
Abstract: By means of a variational approach we find new series representations both for well known mathematical constants, such as $\pi$ and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we have found are all exponentially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar exponentially convergent series for a large class of mathematical functions.
TL;DR: The Riemann Zeta function never vanishes in the region of interest as discussed by the authors, and hence it can never vanish in the whole region of the world, and hence can never be lost.
Abstract: The Riemann Zeta function $\zeta(s)$ never vanishes in the region : $$ \Re s \ge 1- \frac1{5.70176 \log |\Im s|} \quad \quad (|\Im s| \ge 2). $$
TL;DR: In this article, it was shown that the zeta function of an irreducible hypersurface quasi-ordinary singularity f equals the zero function of a plane curve singularity g, and the Puiseux pairs of g can also be recovered from (any set of) distinguished tuples of f.
Abstract: We prove that the zeta function of an irreducible hypersurface quasi-ordinary singularity f equals the zeta function of a plane curve singularity g .I f the local coordinates (x1 ,...,x d+1 )o ff are ‘nice’, then g = f (x1, 0 ,..., 0 ,x d+1). Moreover, the Puiseux pairs of g can also be recovered from (any set of) distinguished tuples of f .
TL;DR: In this article, it was shown that the numerator of the zeta function of a finite flat morphism between two singular reduced irreducible projective projectivealgebraic curves defined over a finite field can be divided into two equal numbers.
Abstract: We prove, as an analogy of a conjecture of Artin, that if
$ Y \rightarrow X $
is a finite flat morphism between two singular reduced absolutely irreducible projective
algebraic curves defined over a finite field, then the numerator of the zeta function
of X divides that of Y
in
$ \mathbb{Z}[T] $
Then, we give some interpretations of this result in terms of
semi-abelian varieties
TL;DR: In this paper, the Euler product expression of the signed double Poisson summation formula and the theory of the double sine function was established for the most basic case by using signed double poisson summations.
Abstract: We construct multiple zeta functions as absolute tensor products of usual zeta functions. The Euler product expression is established for the most basic case by using the signed double Poisson summation formula and the theory of the double sine function.
1 Jun 2004
TL;DR: In this article, an explicit numerical upper bound for the distribution of the zeros of the Riemann zeta function ζ(s) is given for the case where ρ runs over the non-trivial zeros, and the number of zeros ρ = β + iγ of γ ≤ T and 0 < β < 1 by N(T ).
Abstract: We shall give an explicit numerical upper bound for some problem on the distribution of the zeros of the Riemann zeta function ζ(s). It is a continuation of our previous works [5], [8], [9] and [10]. Let ρ run over the non-trivial zeros of ζ(s). We denote the number of the zeros ρ = β + iγ of ζ(s) in 0 < γ ≤ T and 0 < β < 1 by N(T ). If T is not an ordinate of the zeros of ζ(s), let S(T ) denote the value of
TL;DR: In this article, it was shown that large gaps (depending on, and apparently increasing with k) exist between the zeta zeros in Hardy's Z-function and its derivative.
1 Jan 2004
TL;DR: In this article, the authors present a survey of the progress in understanding the distribution of periodic orbits of Anosov flows and hyperbolic flows since the thesis of Margulis was written, focusing on the zeta function techniques introduced by Ruelle and Parry.
Abstract: The purpose of this survey is to outline the progress that has been made in understanding the distribution of periodic orbits of Anosov flows and, more generally, hyperbolic flows since the thesis of Margulis was written. We focus on the zeta function techniques introduced by Ruelle and Parry.
TL;DR: In this article, the volumes, spectra and geodesics of a recently constructed infinite family of five-dimensional inhomogeneous Einstein metrics on the two $S^3$ bundles over two S^2$ fields are examined.
Abstract: The volumes, spectra and geodesics of a recently constructed infinite family of five-dimensional inhomogeneous Einstein metrics on the two $S^3$ bundles over $S^2$ are examined. The metrics are in general of cohomogeneity one but they contain the infinite family of homogeneous metrics $T^{p,1}$. The geodesic flow is shown to be completely integrable, in fact both the Hamilton-Jacobi and the Laplace equation separate. As an application of these results, we compute the zeta function of the Laplace operator on $T^{p,1}$ for large $p$. We discuss the spectrum of the Lichnerowicz operator on symmetric transverse tracefree second rank tensor fields, with application to the stability of Freund-Rubin compactifications and generalised black holes.
Journal Article•
TL;DR: Inequalities involving the Euler zeta function are proved in this article, and applications of the inequalities in estimating the zeta at odd integer values in terms of the known zeta functions at even integer values are discussed.
Abstract: Inequalities involving the Euler zeta function are proved Applications of the inequalities in estimating the zeta function at odd integer values in terms of the known zeta function at even integer values are discussed
TL;DR: The fourth moment of the Riemann zeta function and the second moment of a Maass cusp form were studied in this paper using a construction of Epstein, Hafner and Sarnak.