Showing papers on "Arithmetic zeta function published in 2017"
TL;DR: This article extended the definition of quasi-shuffle algebras so it could be applied to multiple q-zeta values and showed how various algebraic formulas can be obtained in a transparent way.
81 citations
Posted Content•
TL;DR: In this article, a simple variant related to Section 4 in [BW17] leads to the following improvements of Theorem 3 in [ BW17] in [2018] and [2019].
Abstract: This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple variant related to Section 4 in [BW17] leads to the following improvements of Theorem 3 in [BW17]
79 citations
TL;DR: In this paper, the authors established some expressions of series involving harmonic numbers and Stirling numbers of the first kind in terms of multiple zeta values, and presented some new relationships between multiple zero values and multiple zero star values.
54 citations
TL;DR: In this article, two kinds of Euler sums involving generalized harmonic numbers with arbitrary depth were investigated, which established numerous summation formulas including the special values of Arakawa-Kaneno zeta functions and a new formula of multiple zeta values of height one.
Abstract: In this paper, we investigate two kinds of Euler sums that involve the generalized harmonic numbers with arbitrary depth. These sums establish numerous summation formulas including the special values of Arakawa–Kaneno zeta functions and a new formula of multiple zeta values of height one as examples.
41 citations
2 May 2017
41 citations
TL;DR: In this article, generalized harmonic numbers and hyperharmonic numbers under one roof were collected and a unified extension of the Euler sums was presented, where the noninteger property and some arithmetical aspects of this unified extension were considered.
Abstract: Abstract: In this paper we collect two generalizations of harmonic numbers (namely generalized harmonic numbers and hyperharmonic numbers) under one roof. Recursion relations, closed-form evaluations, and generating functions of this unified extension are obtained. In light of this notion we evaluate some particular values of Euler sums in terms of odd zeta values. We also consider the noninteger property and some arithmetical aspects of this unified extension.
34 citations
TL;DR: In this paper, the authors studied meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta function of fractal strings, to arbitrary bounded subsets A of the N-dimensional Euclidean space.
18 citations
TL;DR: In this article, the extended double shuffle relations for interpolated multiple zeta values (MZVs) are established, and a generating function for sums of interpolated MZVs of fixed weight, depth and height is represented by hypergeometric functions.
Abstract: In this paper, the extended double shuffle relations for interpolated multiple zeta values (MZVs) are established. As an application, Hoffman’s relations for interpolated MZVs are proved. Furthermore, a generating function for sums of interpolated MZVs of fixed weight, depth and height is represented by hypergeometric functions, and we discuss some special cases.
17 citations
Posted Content•
TL;DR: In this paper, the authors study motivic zeta functions of degenerating families of Calabi-Yau varieties and show that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois-equivariant Kulikov model.
Abstract: We study motivic zeta functions of degenerating families of Calabi-Yau varieties. Our main result says that they satisfy an analog of Igusa's monodromy conjecture if the family has a so-called Galois-equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture in mirror symmetry.
17 citations
TL;DR: In this article, the authors introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables and systematically investigate such properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results.
Abstract: The main object of this paper is to introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate such its several interesting properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results, and a summation formula. We further consider an application to probability distributions and some important special cases.
TL;DR: In this article, the authors introduce the notion of finite multiple harmonic q-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetrized multiple zero value (SMZV), respectively.
Abstract: We introduce the notion of finite multiple harmonic q-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetrized multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we obtain families of linear relations among these series which induce linear relations among FMZVs and SMZVs of the same form. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic q-series at a primitive root of unity of sufficiently large degree.
TL;DR: In this article, the mixed joint functional independence and generalised universality for a polynomial Euler product and a periodic Hurwitz zeta function, when is a transcendental parameter, are given.
Abstract: Two results related to the mixed joint universality for a polynomial Euler product and a periodic Hurwitz zeta function , when is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.
TL;DR: The fractional derivative of the Dirichlet eta function is computed in order to investigate the behavior of the fractional derivatives of the Riemann zeta function on the critical strip.
Abstract: The fractional derivative of the Dirichlet eta function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function on the critical strip. Its converg ...
TL;DR: In this article, the power series expansions of the function R ( a ) − B (a ) at 0 and at 1 / 2, showed the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so-called Ramanujan constant R( a ) and the beta function B ( a ), 1 − a ).
TL;DR: In this article, the authors investigated some formulae related to Euler Sums and gave the closed-form representations for the sum ∑n≥1Hnn+mmn+kk which can be evaluated in terms of the Riemann zeta functions and generalized harmonic numbers.
Abstract: In this paper, we investigate some formulae related to Euler Sums and give the closed-form representations for the sum ∑n≥1Hnn+mmn+kk which can be evaluated in terms of the Riemann zeta functions and generalized harmonic numbers.
Posted Content•
TL;DR: In this article, the Schur multiple zeta functions of the Euler-Zagier type have been studied and their basic properties including a region of absolute convergence and the case where all variables are the same.
Abstract: We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases.
TL;DR: It is shown that double tails of multiple zeta values satisfy certain recurrence relations and deduce from this a generalization of Euler's classical formula ζ ( 2 ) = 3 ∑ m = 1 ∞ m − 2 ( 2 m m ) − 1, as well as a new and very efficient algorithm for computing these values.
TL;DR: In this paper, it was shown that the Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational, and the same authors also showed that the same result holds for automorphism of groups with diagonal holonomy.
Abstract: We prove that the Reidemeister zeta functions of automorphisms of crystallographic groups with diagonal holonomy $\mathbb{Z}_2$ are rational. As a result, we obtain that Reidemeister zeta functions of automorphisms of almost-crystallographic groups up to dimension 3 are rational.
TL;DR: In this article, the Bernoulli polynomials were used to generate infinite families of linear recurrence relations for the Riemann zeta function at positive even integer arguments.
Posted Content•
28 Mar 2017
TL;DR: In this paper, the authors studied the algebra of elliptic multiple zeta values and proved that the elliptic double shuffle Lie algebra satisfies a double shuffle type family of algebraic relations.
Abstract: We study the algebra $\mathcal{E}$ of elliptic multiple zeta values, which is an elliptic analog of the algebra of multiple zeta values. We identify a set of generators of $\mathcal{E}$, which satisfy a double shuffle type family of algebraic relations, similar to the double-shuffle relations for multiple zeta values. We prove that the elliptic double shuffle relations give all algebraic relations among elliptic multiple zeta values, if (a) the classical double shuffle relations give all algebraic relations among multiple zeta values and if (b) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure.
Posted Content•
TL;DR: In this paper, an explicit formula for a reciprocal sum related to the Riemann zeta function at s = 6 was introduced, and one question related to a computational formula for larger values of s was posed.
Abstract: We introduce an explicit formula for a reciprocal sum related to the Riemann zeta function at s=6, and pose one question related to a computational formula for larger values of s.
TL;DR: In this paper, the analytic continuation of multiple Hurwitz zeta functions was obtained by using a simple and elementary translation formula, and the polar hyperplanes for these functions were located and expressed as coefficients of certain infinite matrices.
Abstract: We obtain the analytic continuation of multiple Hurwitz zeta functions by using a simple and elementary translation formula. We also locate the polar hyperplanes for these functions and express the residues, along these hyperplanes, as coefficients of certain infinite matrices.
TL;DR: In this article, a power series associated with a homogeneous ideal in a polynomial ring was defined, encoding information on the Segre classes defined by extensions of the ideal in projective spaces of arbitrarily high dimension.
1 Aug 2017
TL;DR: In this paper, it was shown that the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and moreover, it also determines their behaviour under local base extensions.
Abstract: Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of the preceding operations can be expressed simultaneously in terms of a single formula, a statement made precise using what we call local maps of Denef type. We show that assuming the existence of such formulae, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.
TL;DR: In this paper, one-parameter series representations for the Riemann zeta function were presented for the following series of series of mathematical constants: Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ζ(3), and π.
Abstract: We present one-parameter series representations for the following series
involving the Riemann zeta function Σ∞n=3 n odd ζ(n)/n sn and Σ∞n=2 n even
ζ(n) n sn and we apply our results to obtain new representations for some
mathematical constants such as the Euler (or Euler-Mascheroni) constant, the
Catalan constant, log 2, ζ(3) and π.
Posted Content•
TL;DR: In this paper, it was shown that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$.
Abstract: We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.
Posted Content•
TL;DR: In this article, the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line were studied and the main object was determining and investigating the density functions of such value-distributions for any algebraic number field.
Abstract: We study the distributions of values of the logarithmic derivatives of the Dedekind zeta functions on a fixed vertical line. The main object is determining and investigating the density functions of such value-distributions for any algebraic number field. We construct the density functions as the Fourier inverse transformations of certain functions represented by infinite products that come from the Euler products of the Dedekind zeta functions.
TL;DR: In this article, an integral representation of the Mathieu $\left(\textbf{a,\bm{\lambda right)$-series is derived by applying the Abel's summation formula.
Abstract: In view of the relationship with the Kr\"{a}tzel function, we derive a new series representation for the $\lambda$-generalized Hurwitz-Lerch Zeta function introduced by H.M. Srivastava [Appl. Math. Inf. Sci. 8 (2014) 1485--1500] and determine the monotonicity of its coefficients. An integral representation of the Mathieu $\left(\textbf{a},\bm{\lambda}\right)$-series is rederived by applying the Abel's summation formula (which provides a slight modification of the result given by Pog\'{a}ny [Integral Transforms Spec. Funct. 16 (8) (2005) 685--689]) and this modified form of the result is then used to obtain a new integral representation for the $\lambda$-generalized Hurwitz-Lerch Zeta function. Finally, by making use of the various results presented in this paper, we establish two sets of two-sided inequalities for the $\lambda$-generalized Hurwitz-Lerch Zeta function.
11 Jan 2017
TL;DR: In this paper, some convexity properties and some inequalities for the (p, k)-analogue of the Gamma function, a (p; k)analogous analogue of the Riemann zeta function,p;k(x) is introduced and some associated inequalities are derived.
Abstract: In this paper, some convexity properties and some inequalities forthe (p; k)-analogue of the Gamma function,a (p; k)-analogue of the celebrated Bohr-Mollerup theorem is given. Furthermore, a (p; k)-analogue of the Riemann zeta function,p;k(x)is introduced andsome associated inequalities are derived. The established results provide the(p; k)-generalizations of some known results concerning the classical Gammafunction