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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.

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 investigate the value-distribution of Epstein zeta-functions $\zeta(s;{\mathcal Q})$, where ${\mathcal Q}$ is a positive definite quadratic form in $n$ variables. We prove an asymptotic formula for the number of $c$-values, i.e., the roots of the equation $\zeta(s;{\mathcal Q})=c$, where $c$ is any fixed complex number. Moreover, we show that, in general, these $c$-values are asymmetrically distributed with respect to the critical line $\operatorname{Re} s=\frac{n}{4}$. This complements previous results on the zero-distribution [30].

This paper discusses the simplest examples of spectral zeta functions, especially those associated with graphs, a subject which has not been much studied. The analogy and the similar structure of these functions, such as their parallel definition in terms of the heat kernel and their functional equations, are emphasized. Another theme is to point out various contexts in which these non-classical zeta functions appear. This includes Eisenstein series, the Langlands program, Verlinde formulas, Riemann hypotheses, Catalan numbers, Dedekind sums, and hypergeometric functions. Several open-ended problems are suggested with the hope of stimulating further research.

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.

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.

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.

The paper presents formulae for certain series involving the Riemann zeta function. These formulae are generalizations, in a natural way, of well known formulae, originating from Leonhard Euler. Formulae that existed only for initial values n = 0 , 1 are now found for every natural n . Relevant connections with various known results are also pointed out.

Assume the Riemann Hypothesis. We establish a local structure theorem for zeros of the Riemann zeta-function ζ ( s ) and its derivative ζ ′ ( s ) . As an application, we prove a stronger form of half of a conjecture of Radziwill [18] concerning the global statistics of these zeros. Roughly speaking, we show that on the Riemann Hypothesis, if there occurs a small gap between consecutive zeta zeros, then there is exactly one zero of ζ ′ ( s ) lying not only very close to the critical line but also between that pair of zeta zeros. This refines a result of Zhang [22] . Some related results are also shown. For example, we prove a weak form of a conjecture of Soundararajan, and suggest a repulsion phenomena for zeros of ζ ′ ( s ) .

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.

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

The Hasse–Witt matrix of a hypersurface in P n over a finite field of characteristic p gives essentially complete mod p information about the zeta function of the hypersurface. But if the degree d of the hypersurface is ≤n, the zeta function is trivial mod p and the Hasse–Witt matrix is zero-by-zero. We generalize a classical formula for the Hasse–Witt matrix to obtain a matrix that gives a nontrivial congruence for the zeta function for all d. We also describe the differential equations satisfied by this matrix and prove that it is generically invertible.

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.

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.

In this paper, we give a definition of the p-adic Arakawa–Kaneko–Hamahata zeta functions. These zeta functions interpolate Hamahata's poly-Euler polynomials at non-positive integers. We prove the derivative formula, the difference equation and the reflection formula of these zeta functions. Furthermore, we also prove a sums of products identity and a closed form of Hamahata's poly-Euler polynomials in terms of the Stirling numbers of the second kind.

On a certain probability space, an analytic random element and a random variable both related to the Riemann zeta function and a measurable measure preserving transformation are considered. For these entities, an equality generalizing the classical ergodic Birkhoff–Khinchine theorem is proved.

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.

In this note, we build on recent work in [7] to establish formulas for ζ(2n) as sums over all the unrestricted integer partitions of n.

The authors have previously reported the existence of a morphism between the Riemann zeta function and the “Cole and Cole” canonical transfer functions observed in dielectric relaxation, electrochemistry, mechanics and electromagnetism. The link with self-similar structures has been addressed for a long time and likewise the discovered of the incompleteness which may be attached to any dynamics controlled by non-integer derivative operators. Furthermore it was already shown that the Riemann Hypothesis can be associated with a transition of an order parameter given by the geometric phase attached to the fractional operators. The aim of this note is to show that all these properties have a generic basis in category theory. The highlighting of the incompleteness of non-integer operators considered as critical by some authors is relevant, but the use of the morphism with zeta function reduces the operational impact of this issue without limited its epistemological consequences.

We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends previous work of ours to all curves for which a good lift to characteristic zero is known. We develop all the necessary bounds, analyse the complexity of the algorithm and provide a complete implementation. publisher: Elsevier articletitle: Counting points on curves using a map to P1, II journaltitle: Finite Fields and Their Applications articlelink: http://dx.doi.org/10.1016/j.ffa.2016.12.008 content_type: article copyright: © 2017 Elsevier Inc. All rights reserved. ispartof: Finite Fields and their Applications vol:45 pages:301-322 status: published

In this paper, we introduce new sequences, which generalize the celebrated DeTemple sequence, having enhanced speed of convergence. We also give a new representation for Euler’s constant in terms of the Riemann zeta function evaluated at positive odd integers.

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 ...

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.

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.

The authors present the power series expansions of the function R ( a ) − B ( a ) at a = 0 and at a = 1 / 2 , show 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 ) ≡ B ( a , 1 − a ) , and obtain asymptotically sharp lower and upper bounds for R ( a ) in terms of B ( a ) and polynomials. In addition, some properties of the Riemann zeta function ζ ( n ) , n ∈ N , and its related sums are derived.

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 consider a family of approximations of the Dedekind zeta function ζK(s) of a number field K/Q. Weighted L^2-norms of the difference of two such approximations of ζK(s) are computed. We work with a weight which is a compactly supported smooth function. Mean square estimates for the difference of approximations of ζK(s) can be obtained from such weighted L^2-norms. Some results on the location of zeros of a family of approximations of Dedekind zeta functions are also derived. These results extend results of Gonek and Montgomery on families of approximations of the Riemann zeta-function.

The objective of this dissertation is to study the Riemann zeta function in particular it will examine its analytic continuation, functional equation and applications. We will begin with some historical background, then define of the zeta function and some important tools which lead to the functional equation. We will present four different proofs of the functional equation. In addition, the (s) has generalizations, and one of these the Dirichlet L-function will be presented. Finally, the zeros of (s) will be studied.

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.

In this paper, we study factorizability of \(\mathbb{C}\)-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) "non-factorizability" of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function.

In this paper, we use the generating function of the Bernoulli polynomials to introduce a number of infinite families of linear recurrence relations for the Riemann zeta function at positive even integer arguments, ζ ( 2 n ) .

We calculate a certain mean-value of meromorphic functions by using specific ergodic transformations, which we call affine Boolean transformations. We use Birkhoff's ergodic theorem to transform the mean-value into a computable integral which allows us to completely determine the mean-value of this ergodic type. As examples, we introduce some applications to zeta functions and L-functions. We also prove an equivalence of the Lindelof hypothesis of the Riemann zeta function in terms of its certain ergodic value distribution associated with affine Boolean transformations.

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 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)

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.

Assuming a weaker form of the Riemann hypothesis for Dedekind zeta functions by allowing Siegel zeros, we extend a classical result of Cramer on the number of primes in short intervals to prime ideals of the ring of integers in cyclotomic extensions with norms belonging to such intervals. The extension is uniform with respect to the degree of the cyclotomic extension. Our approach is based on the arithmetic of cyclotomic fields and analytic properties of their Dedekind zeta functions together with a lower bound for the number of primes over progressions in short intervals subject to similar assumptions. Uniformity with respect to the modulus of the progression is obtained and the lower bound turns out to be best possible, apart from constants, as shown by the Brun–Titchmarsh theorem.

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.

In this paper, we first generalize the Kronecker limit formula for a class of Epstein zeta functions using new approximation formulas. This enables us to derive some applications to the class number of quadratic imaginary number fields K and the period ratios of elliptic curves with complex multiplication.

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