Search

In [1] the author proposed two new results concerning the prime zeta function and the Riemann zeta function but they turn out to be wrong. In the present paper we provide their correct form.

The values at 1 of single-valued multiple polylogarithms span a certain subalgebra of multiple zeta values. The properties of this algebra are studied from the point of view of motivic periods.

We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad categories of errors. First, we consider some significant errors made by highly-regarded mathematicians. In some cases these errors were not detected until many years after their publication. Second, we consider in some detail an error that was recently detected by the author. This error in a refereed journal led to further errors by at least one author who relied on the (incorrect) result. Finally, we mention some instructive errors that have been detected in the author's own published papers., Comment: 25 pages, 77 references. References updated in v3

In this paper we introduce and study double tails of multiple zeta values. We show, in particular, that they 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 to all multiple zeta values, as well as a new and very efficient algorithm for computing these values.

We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical "repulsion" between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function's magnitude on the 1-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related L-functions., Comment: Fixed more typos and minor corrections

Multizeta values for F q [ θ ] were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In this present paper we give certain depth 2 families of zeta-like multizeta values, namely those whose ratio to the zeta value of the same weight is rational.

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.

The paper summarizes the computation results pursuing peak values of the Riemann zeta function. The computing method is based on the RS-Peak algorithm by which we are able to solve simultaneous Diophantine approximation problems efficiently. The computation environment was served by the SZTAKI Desktop Grid operated by the Laboratory of Parallel and Distributed Systems at the Hungarian Academy of Sciences and the ATLAS supercomputing cluster of the Eotvos Lorand University, Budapest. We present the largest Riemann zeta value known till the end of 2016.

We consider a broad family of zeta functions which includes the classical zeta functions of Riemann and Hurwitz, the beta and eta functions of Dirichlet, and the Lerch transcendent, as well as the Arakawa–Kaneko zeta functions and the recently introduced alternating Arakawa–Kaneko zeta functions. We construct their [Formula: see text]-adic analogues and indicate the many strong connections between the complex and [Formula: see text]-adic versions. As applications, we focus on the alternating case and show how certain families of alternating odd harmonic number series can be expressed in terms of Riemann zeta and Dirichlet beta values.

In an attempt to improve Ramanujan's unsuccessful evaluation of double sums on Hurwitz zeta functions, we introduce more general multiple zeta values on Hurwitz zeta functions defined as ∑k1=0∞∑k2=0∞⋯∑kr=0∞(k1+x1)−α1×[(k1+x1)+(k2+x2)]−α2×⋯×[(k1+x1)+(k2+x2)+⋯+(kr+xr)]−αr, with α1,α2,…,αr positive integers, αr≥2 and positive numbers x1,x2,…,xr. Especially, we extend Euler decomposition theorem which expressed a product of two Riemann zeta values in terms of Euler double sums, to a more general decomposition theorem which expressed products of n Hurwitz zeta values in terms of multiple zeta values on Hurwitz zeta functions as mentioned before. Furthermore, we apply various differential operators to the resulted decomposition theorem to produce more decomposition theorems concerning products of multiples of values of Hurwitz zeta function.

Recently various extensions of Hurwitz-Lerch Zeta functions have been investigated. Here, we first introduce a further generalization of the extended Hurwitz-Lerch Zeta functions. Then we investigate certain interesting and (potentially) useful properties, systematically, of the generalization of the extended Hurwitz-Lerch Zeta functions, for example, various integral representations, Mellin transform, generating functions and extended fractional derivatives formulas associated with these extended generalized Hurwitz-Lerch Zeta functions. An application to probability distributions is further considered. Some interesting special cases of our main results are also pointed out.

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 ...

Recently, the first author has extended the definition of the zeta function associated with fractal strings to arbitrary bounded subsets $A$ of the $N$-dimensional Euclidean space ${\mathbb R}^N$, for any integer $N\ge1$. It is defined by $\zeta_A(s)=\int_{A_{\delta}}d(x,A)^{s-N}\,\mathrm{d} x$ for all $s\in\mathbb{C}$ with $\operatorname{Re}\,s$ sufficiently large, and we call it the distance zeta function of $A$. Here, $d(x,A)$ denotes the Euclidean distance from $x$ to $A$ and $A_{\delta}$ is the $\delta$-neighborhood of $A$, where $\delta$ is a fixed positive real number. We prove that the abscissa of absolute convergence of $\zeta_A$ is equal to $\overline\dim_BA$, the upper box (or Minkowski) dimension of $A$. Particular attention is payed to the principal complex dimensions of $A$, defined as the set of poles of $\zeta_A$ located on the critical line $\{\mathop{\mathrm{Re}} s=\overline\dim_BA\}$, provided $\zeta_A$ possesses a meromorphic extension to a neighborhood of the critical line. We also introduce a new, closely related zeta function, $\tilde\zeta_A(s)=\int_0^{\delta} t^{s-N-1}|A_t|\,\mathrm{d} t$, called the tube zeta function of $A$. Assuming that $A$ is Minkowski measurable, we show that, under some mild conditions, the residue of $\tilde\zeta_A$ computed at $D=\dim_BA$ (the box dimension of $A$), is equal to the Minkowski content of $A$. More generally, without assuming that $A$ is Minkowski measurable, we show that the residue is squeezed between the lower and upper Minkowski contents of $A$. We also introduce transcendentally quasiperiodic sets, and construct a class of such sets, using generalized Cantor sets, along with Baker's theorem from the theory of transcendental numbers., Comment: 54 pages, corrected misprints, reduced number of self-citations

We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(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, ?(3) and ?.

We obtain new estimates for the maximum and minimum of the argument of the Riemann zetafunction on very short segments of the critical line. These results are based on the Riemann hypothesis.

We show for all $1/2 \le \sigma \le 1$ and $t\ge 3$ that $\zeta(\sigma+it)| \le 76.2 t^{4.45 (1-\sigma)^{3/2}}$, where $\zeta$ is the Riemann zeta function. This significantly improves the previous bounds, where $4.45$ is replaced by $18.8$. New ingredients include a method of bounding $\zeta(s)$ in terms of bounds for Vinogradov's Integral (aka Vinogradov's Mean Value) together with bounds for "incomplete Vinogradov systems", explicit bounds for Vinogradov's integral which strengthen slightly bounds of Wooley (Mathematika 39 (1992), no. 2, 379-399), and explicit bounds for the count of solutions of "incomplete Vinogradov systems", following ideas of Wooley (J. Reine Angew. Math. 488 (1997), 79-140), Comment: Published in 2002, Proc. London Math. Soc. This version corrects small typos on 5 pages of the published version (a list can be found on the author's web page)

Fyodorov, Hiary & Keating established an intriguing connection between the maxima of log-correlated processes and the ones of the Riemann zeta function on a short interval of the critical line. In particular, they suggest that the analogue of the free energy of the Riemann zeta function is identical to the one of the Random Energy Model in spin glasses. In this paper, the connection between spin glasses and the Riemann zeta function is explored further. We study a random model of the Riemann zeta function and show that its two-overlap distribution corresponds to the one of a one-step replica symmetry breaking (1-RSB) spin glass. This provides evidence that the local maxima of the zeta function are strongly clustered.

In this article we obtain the meromorphic continuation of multiple zeta functions, together with a complete list of their poles and residues, by means of an elementary and simple translation formula for these multiple zeta functions. The use of matrices to express this translation formula leads, in particular, to a succinct description of the residues of the multiple zeta functions. We conclude our paper by introducing certain interesting weighted variants of multiple zeta functions. They are shown to behave particularly nicely with respect to product formulas and location of poles.

Two results related to the mixed joint universality for a polynomial Euler product $\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function $\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$, when $\unicode[STIX]{x1D6FC}$ 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.

We introduce and study an analogue of the cotangent zeta function in the function fields setting. We establish a relation of our newly introduced zeta function to the Apostol–Dedekind sum, and then prove a functional equation for our function. Finally, we compute special values of the cotangent zeta function at quadratic irrationals.

Following Bachmann's recent work on bi-brackets and multiple Eisenstein series, Zudilin introduced the notion of multiple q-zeta brackets, which provides a q-analog of multiple zeta values possessing both shuffle as well as quasi-shuffle relations. The corresponding products are related in terms of duality. In this work we study Zudilin's duality construction in the context of classical multiple zeta values as well as various q-analogs of multiple zeta values. Regarding the former we identify the derivation relation of order two with a Hoffman-Ohno type relation. Then we describe relations between the Ohno-Okuda-Zudilin q-multiple zeta values and the Schlesinger-Zudilin q-multiple zeta values., revised version, accepted for publication in Advances in Mathematics

Given an abelian Galois extension K / F of number fields, a quaternion algebra A over F that is ramified at all infinite primes, and a character χ of the Galois group of K over F, we consider the twist of the zeta function of A by the character χ. We show that such twisted zeta functions provide a factorization of the zeta function of A ( K ) = A ⊗ F K . Also, the quotient of the zeta function for A ( K ) by the zeta function for A is entire when K / F is Galois of odd degree, not necessarily abelian. Examples show that the hypothesis of odd degree here may be relaxed, but not eliminated. So the analogue of the Aramata–Brauer holomorphy theorem in this situation requires some restriction. A key step in the proof of this analogue for K / F Galois of odd degree consists in showing that if a finite prime of K contains a ramified prime for A over F, then it is ramified for A ( K ) over K.

We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an arbitrary number field~$K$, at the rational primes which are non-split in~$K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime., Comment: 19 pages; to appear in Israel J. Math

This work is dedicated to the promotion of the results C. Muntz obtained modifying zeta functions. The properties of zeta functions are studied; these properties lead to new regularities of zeta functions. The choice of a special type of modified zeta functions allows estimating the Riemann’s zeta function and solving Riemann Problem-Millennium Prize Problem.

The zeta function associated with the category of finite Abelian groups is defined and studied. This zeta function is expressed in terms of the Riemann zeta function.

This paper is the first in a series of four devoted to the abc conjecture, the Rie-mann Hypothesis, Fermat-Wiles Theorem and its extensions, Fermat Numbers and their generalizations. We propose here a method of solving the conjecture abc based on 3 well-established properties of numbers : the sequence of successive odd numbers up to a given bound, the property of the logarithmic function in the interval ]0,1[ and the condition necessary to talk about the conjecture. This method offers the following triple interest : it solves the abc conjecture by determining the associated constant ; it solves a stronger version of the representation of even numbers as the sum which implies the Goldbach conjecture and the second Hardy-Littlewood conjecture ; it elaborates the essential elements for the demonstration of Riemann Hypothesis, of Lemoine's conjecture, and of twin primes conjectures for which it provides a lower bound on π 2. This method makes it possible to integrate into a single a large set of problems including the abc conjecture and the Riemann Hypothesis. This is matter to follow.

This paper is the first in a series of four devoted to the abc conjecture, the Rie-mann Hypothesis, Fermat-Wiles Theorem and its extensions, Fermat Numbers and their generalizations. We propose here a method of solving the conjecture abc based on 3 well-established properties of numbers : the sequence of successive odd numbers up to a given bound, the property of the logarithmic function in the interval ]0,1[ and the condition necessary to talk about the conjecture. This method offers the following triple interest : it solves the abc conjecture by determining the associated constant ; it solves a stronger version of the representation of even numbers as the sum which implies the Goldbach conjecture and the second Hardy-Littlewood conjecture ; it elaborates the essential elements for the demonstration of Riemann Hypothesis, of Lemoine's conjecture, and of twin primes conjectures for which it provides a lower bound on π 2. This method makes it possible to integrate into a single a large set of problems including the abc conjecture and the Riemann Hypothesis. This is matter to follow.

Producción Científica, 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, Ministerio de Economía, Industria y Competitividad ( grants MTM2012-36917-C03-03 / MTM2015-65764- C3-2-P)

The zeta Mahler measure is the generating function of higher Mahler measures. In this article, explicit formulas of higher Mahler measures, and relations between higher Mahler measures and multiple zeta (star) values are showed by observing connections between zeta Mahler measures and the generating functions of multiple zeta (star) values. Additionally, connections between higher Mahler measures and Dirichlet L-values associated with primitive quadratic characters are discussed.

This paper extends tools developed by Crandall (2012) 16 to provide robust, high-precision methods for computation of the incomplete Gamma function and the Lerch transcendent. We then apply these to the corresponding computation of the Hurwitz zeta function and so of Dirichlet L-series and character polylogarithms.

Text The Arakawa–Kaneko zeta functions interpolate the poly-Bernoulli polynomials at the negative integers, while their values at the positive integers are connected to multiple zeta values and harmonic number sums. Here we construct p-adic analogues of these functions and show that these complex formulas have good p-adic counterparts. The method of construction, summing a series of forward differences of p-adic power functions, is a p-adic adaptation of an everywhere-convergent series for the Hurwitz zeta function due to Hasse. We also apply this method to construct a p-adic Lerch transcendent. Expressions for zeta values and related constants as harmonic number series which converge in both complex and p-adic senses are a prominent feature of this approach. Video For a video summary of this paper, please visit http://youtu.be/Mqd6W0TW9D0 .

We present a new method for fast approximation of zeta constants, i.e., values ?(n) of the Riemann zeta function, n ? 2, n is an integer, by rational fractions. The method makes it possible to fast approximate zeta constants and certain combinations of successive values of zeta constants by rational fractions. By choosing values of coefficients involved in the combinations, one can control the convergence rate of the approximations and the computation complexity for the zeta constants.

Conjecturally, almost all graphs are determined by their spectra. This problem has also been studied for variants such as the spectra of the Laplacian and signless Laplacian. Here we consider the problem of determining graphs with Ihara and Bartholdi zeta functions, which are also computable in polynomial time. These zeta functions are geometrically motivated, but can be viewed as certain generalizations of characteristic polynomials. After discussing some graph properties determined by zeta functions, we show that large classes of cospectral graphs can be distinguished with zeta functions and enumerate graphs distinguished by zeta functions on $\le 11$ vertices. This leads us to conjecture that almost all graphs which are not determined by their spectrum are determined by zeta functions. Along the way, we make some observations about the usual types of spectra and disprove a conjecture of Setyadi and Storm about Ihara zeta functions determining degree sequences., Comment: 23 pages, to appear in Linear Algebra and its Applications, minor revisions and strengthened results in Section 3.2

We find the sum of series of the form $$\sum\limits_{i = 1}^\infty {\frac{{f(i)}}{{{i^r}}}} $$ for some special functions f. The above series is a generalization of the Riemann zeta function. In particular, we take f as some values of Hurwitz zeta functions, harmonic numbers, and combination of both. These generalize some of the results given in Mező’s paper (2013). We use multiple zeta theory to prove all results. The series sums we have obtained are in terms of Bernoulli numbers and powers of π.

We find a Selberg zeta function expression of certain one-loop spin partition functions on three-dimensional thermal anti-de Sitter space. Of particular interest is the partition function of higher spin fermionic particles. We also set up, in the presence of spin, a Patterson-type formula involving the logarithmic derivative of zeta.

The classical Euler decomposition theorem expresses a product of two Riemann zeta values in terms of double Euler sums. Also, the sum formula expresses a single Riemann zeta value in terms of multiple zeta values of depth n. In this paper, we express a product of n Riemann zeta values in terms of multiple zeta values of depth n. We establish a new and general explicit formula ζ(d1+2)ζ(d2+2)⋯ζ(dn+2)=∑|α|=|d|+2n−1ζ(α1,α2,…,αn+1)∑σ∈Snσd{(hndn+1)(hn−1dn−1+1)⋯(h2d2+1)}, where hj=dj+2+∑l≥j(αl−dl−2), Sn is the symmetric group of n objects, and σd is the induced permutation of σ on the set {d1,d2,…,dn}. The case n=2 gives the Euler decomposition theorem.

In this paper, we provide an asymptotic formula for the higher derivatives of the Hurwitz zeta function with respect to its first argument that does not need recurrences. As a by-product, we correct some formulas that have appeared in the literature.

We look at two analogs each for the well-known congruences of Fermat and Wilson in the case of polynomials over finite fields. When we look at them modulo higher powers of primes, we find interesting relations linking them together, as well as linking them with derivatives and zeta values. The link with the zeta value carries over to the number field case, with the zeta value at 1 being replaced by Euler's constant.

We establish a formula for the volume Poincaré series of a log smooth scheme. This yields in particular a new expression and a smaller set of candidate poles for the motivic zeta function of a hypersurface singularity and of a degeneration of Calabi–Yau varieties. ispartof: Comptes Rendus. Mathématique vol:353 issue:3 pages:261-264 status: published