Showing papers on "Arithmetic zeta function published in 2018"
TL;DR: In this article, the authors studied the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd-dimensional manifold.
Abstract: In this paper we study the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd-dimensional manifold. These are functions of a complex variable s in some right half-plane of $$\mathbb {C}$$
. Using the Selberg trace formula for arbitrary finite dimensional representations of the fundamental group of the manifold, we establish the meromorphic continuation of the dynamical zeta functions to the whole complex plane. We explicitly describe the singularities of the Selberg zeta function in terms of the spectrum of certain twisted Laplace and Dirac operators.
21 citations
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TL;DR: In this paper, the Fourier expansion method was used to evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions, and relations between the values of a class of the Hurwitz type (or Lerch-type) Euler Zeta functions at rational arguments were given.
Abstract: The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} \zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $\zeta_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.
19 citations
TL;DR: By the use of Hermite-Hadamard inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given in this article.
Abstract: By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is...
6 citations
TL;DR: In this article, an infinite sequence of inequalities involving the Riemann zeta function with even arguments ζ (2n) and the Chebyshev-Stirling numbers of the first kind was given.
Abstract: In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments ζ (2n) and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions [18]. An interesting asymptotic formula related to the n th complete homogeneous symmetric function is conjectured in this context:
4 citations
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TL;DR: A survey on weight enumerators, zeta functions and Riemann hypothesis for linear and algebraic-geometry codes is given in this article, where the authors also present a survey on the Riemans hypothesis for algebraic geometry codes.
Abstract: This is a survey on weight enumerators, zeta functions and Riemann hypothesis for linear and algebraic-geometry codes.
4 citations
TL;DR: In this paper, the duality relation for double zeta values and the sum of multiple zeta value whose first components are 2's are deduced from the derivation relation, which is known as a subclass of the extended double shuffle relation.
Abstract: We consider the problem of deducing the duality relation from the extended double shuffle relation for multiple zeta values. Especially we prove that the duality relation for double zeta values and that for the sum of multiple zeta values whose first components are 2’s are deduced from the derivation relation, which is known as a subclass of the extended double shuffle relation.
4 citations
TL;DR: In this paper, the uniqueness problem of Riemann zeta functions was studied and it was shown that the RiemANN zeta function is uniquely determined in terms of the preimages of three complex values a,b,0 except possibly a set G with n(r,G)=o(r), where G is called an exceptional set.
Abstract: The paper concerns the uniqueness problem of Riemann zeta-function. It is showed that the Riemann zeta-function is uniquely determined in terms of the preimages of three complex values a,b,0 except possibly a set G with n(r,G)=o(r), where G is called an exceptional set.
3 citations
TL;DR: In this paper, the authors introduce a multiple zeta function as a generalization of the Mordell-Tornheim zeta functions and study its analytic properties, revealing its behavior at non-positive arguments by a new method.
Abstract: In this paper, we introduce a certain multiple zeta function as a generalization of the Mordell–Tornheim zeta function and study its analytic properties. In particular, we reveal its behavior at non-positive arguments by a new method which may be useful in a more general setting.
2 citations
TL;DR: In this paper, a brief overview on zeta functions of curve singularities is given and some evidences on how these and global zeta function associated to singular algebraic curves over perfect fields relate to each other.
Abstract: The purpose of this note is to give a brief overview on zeta functions of curve singularities and to provide some evidences on how these and global zeta functions associated to singular algebraic curves over perfect fields relate to each other.
1 citations
TL;DR: In this paper, the authors give certain depth 2 families of zeta-like multizeta values, namely those whose ratio to the zeta value of the same weight is rational.
1 citations
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TL;DR: The convergence of a sequence of Cauchy sequences is conjectured in this article, which if shown to be true, would prove the Riemann hypothesis by way of LeClair and Franca's transcendental equation criteria.
Abstract: The convergence of a sequence of Cauchy sequences is conjectured; which if shown to be true, would prove the Riemann hypothesis by way of LeClair and Franca's transcendental equation criteria.
TL;DR: In this paper, an asymptotic relationship between the product of fractional parts with powers of integers and the values of the Riemann zeta function has been shown.
Abstract: A well-known result, due to Dirichlet and later generalized by de la Vallee–Poussin, expresses a relationship between the sum of fractional parts and the Euler–Mascheroni constant. In this paper, we prove an asymptotic relationship between the summation of the products of fractional parts with powers of integers on the one hand, and the values of the Riemann zeta function, on the other hand. Dirichlet’s classical result falls as a particular case of this more general theorem.
30 Jun 2018
Abstract: We propose and develop yet another approach to the problem of summation of series involving the Riemann Zeta function , the Hurwitz generalized Zeta function , the Polygamma function , and the polylogarithmic function. The key ingredients in our approach include certain known integral representations for the Zeta function.. The method developed in this paper is illustrated by numerous examples of closed-form evaluations of series of the aforementioned types. Many of the resulting summation formulas are believed to be new. The method has been implemented in Mathematica.
30 Jun 2018
Abstract: We propose and develop yet another approach to the problem of summation of series involving the Riemann Zeta function , the Hurwitz generalized Zeta function , the Polygamma function , and the polylogarithmic function. The key ingredients in our approach include certain known integral representations for the Zeta function.. The method developed in this paper is illustrated by numerous examples of closed-form evaluations of series of the aforementioned types. Many of the resulting summation formulas are believed to be new. The method has been implemented in Mathematica.
TL;DR: This work defines the rank metric zeta function of a code as a generating function of its normalized q-binomial moments and shows invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zetafunction for a family of codes.
Abstract: We define the rank metric zeta function of a code as a generating function of its normalized q-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.