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In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus. [ABSTRACT FROM AUTHOR]

Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose–Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations. [ABSTRACT FROM AUTHOR]

Numerous logarithmic integrals have been extensively documented in the literature. This paper presents an algorithmic evaluation of a specific class of these integrals. Our systematic approach, rooted in logarithmic principles, enables us to extend our findings to other cases within this family of integrals. Furthermore, we explore special cases derived from our main results, thereby enhancing the applicability and significance of our work for a wider audience of researchers. [ABSTRACT FROM AUTHOR]

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C 2 defined by means of the Epstein ζ (s ; Q) and Hurwitz ζ (s , α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

Yamamoto’s integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto’s integral. We determine the value of infinite series of the form: ∑ m1,...,mn,ℓ1,...,ℓk≥1 H(a1)m1 · · · H(an)mn/mb11 · · · mbnn ℓc11 · · · ℓckk (m1+···+mn+ℓ1+···+ℓkℓk), in terms of a finite sum of multiple zeta values, for positive integers a1, . . ., an, b1, . . ., bn, c1, . . ., ck. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problem posed by Kuba by expressing ∑ j , k ≥ 1 H k (u) H j (v) H j + k (w) j r k s (j + k) t as a linear combination of multiple zeta values. These sums include Tornheim's double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series. [ABSTRACT FROM AUTHOR]

In this paper, we derive the asymptotic formulas B*r,s,t(n) such that ... where Re(r + s) > 1 and t < C. It is evident that the asymptotic formulas for the inverses of the tails of both the Riemann zeta function and the Hurwitz zeta function on the half-plane Re(s) > 1 are its corollaries. Subsequently we provide the asymptotic formulas for the Riemann zeta function and the Hurwitz zeta function on the half-plane Re(s) < 0. Finally, we study the asymptotic formulas of the inverse of the tails of the Dirichlet L-function for Re(s) > 1 and Re(s) < 0. [ABSTRACT FROM AUTHOR]

Several extensions of the basic scalar variable logistic density to the multivariate and matrix-variate cases, in the real and complex domains, are given where the extended forms end up in extended zeta functions. Several cases of multivariate and matrix-variate Bayesian procedures, in the real and complex domains, are also given. It is pointed out that there are a range of applications of Gaussian and Wishart-based matrix-variate distributions in the complex domain in multi-look data from radar and sonar. It is hoped that the distributions derived in this paper will be highly useful in such applications in physics, engineering, statistics and communication problems, because, in the real scalar case, a logistic model is seen to be more appropriate compared to a Gaussian model in many industrial applications. Hence, logistic-based multivariate and matrix-variate distributions, especially in the complex domain, are expected to perform better where Gaussian and Wishart-based distributions are currently used. [ABSTRACT FROM AUTHOR]

In this paper, we continue the study of efficient algorithms for the computation of the Riemann zeta function on the complex plane. We introduce two precalculated arrays-based modifications of MB-method. We perform numerical experiments with these algorithms using Zetafast as a benchmark and apply the algorithms for the visualizations of fractal structures associated with the Riemann zeta function. [ABSTRACT FROM AUTHOR]

This paper considers a generalization of the Rényi theorem to the case of a structural distribution with a scale parameter. In terms of the zeta metric, some estimates of the convergence rate in the generalized Rényi theorem are obtained when the structural mixed Poisson distribution of the summation index is a scale mixture of the generalized gamma distribution. Estimates of the convergence rate for the structural digamma distribution are given as a special case. The paper extends the results previously obtained for the generalized gamma distribution. [ABSTRACT FROM AUTHOR]

In certain problems in model building and Bayesian analysis, the results end up in forms connected with generalized zeta functions. This necessitates the introduction of an extended form of the generalized zeta function. Such an extended form of the zeta function is introduced in this paper. In model building situations and in various types of applications in physical, biological and social sciences and engineering, a basic model taken is the Gaussian model in the univariate, multivariate and matrix-variate situations. A real scalar variable logistic model behaves like a Gaussian model but with a thicker tail. Hence, for many of industrial applications, a logistic model is preferred to a Gaussian model. When we study the properties of a logistic model in the multivariate and matrix-variate cases, in the real and complex domains, invariably the problem ends up in the extended zeta function defined in this paper. Several such extended logistic models are considered. It is also found that certain Bayesian considerations also end up in the extended zeta function introduced in this paper. Several such Bayesian models in the multivariate and matrix-variate cases in the real and complex domains are discussed. It is stated in a recent paper that "Quantum Mechanics is just the Bayesian theory generalized to the complex Hilbert space". Hence, the models developed in this paper are expected to have applications in quantum mechanics, communication theory, physics, statistics and related areas. [ABSTRACT FROM AUTHOR]

Matrix-variate Gaussian-type or Wishart-type distributions in the real domain are widely used in the literature. When the exponential trace has an arbitrary power and when the factors involving a determinant and a trace enter into the model or a matrix-variate gamma-type or Wishart-type model with an exponential trace having an arbitrary power, they are extremely difficult to handle. One such model with factors involving a trace and a determinant and the exponential trace having an arbitrary power, in the real domain, is known in the literature as the Kotz model. No explicit evaluation of the normalizing constant in the Kotz model seems to be available. The normalizing constant that is widely used in the literature, is interpreted as the normalizing constant in the general model, and that is referred to as a Kotz model does not seem to be correct. One of the main contributions in this paper is the introduction of matrix-variate distributions in the real and complex domains belonging to the Gaussian-type, gamma-type, and type 1 and type 2 beta-types, or Mathai's pathway family, when the exponential trace has an arbitrary power and explicit evaluations of the normalizing constants therein. All of these models are believed to be new. Another new contribution is the logistic-based extensions of the models in the real and complex domains, with the exponential trace having an arbitrary exponent and connecting to extended zeta functions introduced by this author recently. The techniques and steps used at various stages in this paper will be highly useful for people working in multivariate statistical analysis, as well as for people applying such models in engineering problems, communication theory, quantum physics, and related areas, apart from statistical applications. [ABSTRACT FROM AUTHOR]

This paper examines the mysterious non-trivial zeros of the Riemann zeta function ... and explains their role, e.g., in the computation of the error term in Riemann's J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ... and the prime numbers. [ABSTRACT FROM AUTHOR]

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler's identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ℓ p → ℓ q bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of generalized Krätzel functions, which are the kernel functions of type-1 and type-2 pathway transforms. Various analytical properties such as Lipschitz continuity, fixed point property and integrability of these functions are investigated. Furthermore, the paper introduces two new inequalities associated with generalized Krätzel functions. The composition formulae for various fractional operators, such as Grünwald-Letnikov fractional differential operator, Riemann-Liouville fractional integral and differential operators with these functions are also obtained. Additionally, it has been demonstrated that these kernel functions are related to the Heaviside step function, the Dirac delta function, and the Riemann zeta function. A computable series representation of the kernel function is also obtained. Further scope of the work is pointed out by modifying the kernel function with the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

In the paper, we consider simultaneous approximation of a pair of analytic functions by discrete shifts ζuN (s + ikh1; a) and ζuN (s + ikh2, α; b) of the absolutely convergent Dirichlet series connected to the periodic zeta-function with multiplicative sequence a, and the periodic Hurwitz zeta-function, respectively. We suppose that uN → ∞ and uN ≪ N2 as N → ∞, and the set {(h1 log p: p ∈ P), (h2 log(m + α): m ∈ N0), 2π} is linearly independent over Q. [ABSTRACT FROM AUTHOR]

In the paper, we prove a limit theorem in the sense of the weak convergence of probability measures for the modified Mellin transform Z (s) , s = σ + i t , with fixed 1 / 2 < σ < 1 , of the square | ζ (1 / 2 + i t) | 2 of the Riemann zeta-function. We consider probability measures defined by means of Z (σ + i φ (t)) , where φ (t) , t ⩾ t 0 > 0 , is an increasing to + ∞ differentiable function with monotonically decreasing derivative φ ′ (t) satisfying a certain normalizing estimate related to the mean square of the function Z (σ + i φ (t)) . This allows us to extend the distribution laws for Z (s) . [ABSTRACT FROM AUTHOR]

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel's original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that "half" of Siegel's zeta functions correspond to holomorphic modular forms. [ABSTRACT FROM AUTHOR]

In this paper we obtain sufficient condition for univalence of analytic functions defined by Hurwitz-Lerach Zeta function. [ABSTRACT FROM AUTHOR]

Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) approximating a given analytic function defined on a certain strip σ ^ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied. [ABSTRACT FROM AUTHOR]

In this paper, the approximation of analytic functions by shifts  ζ P (s + i τ)  of Beurling zeta-functions  ζ P (s)  of certain systems  P  of generalized prime numbers is discussed. It is required that the system of generalized integers  N P  generated by  P  satisfies  ∑ m ⩽ x , m ∈ N 1 = a x + O (x δ) ,  a > 0 ,  0 ⩽ δ < 1 , and the function  ζ P (s)  in some strip lying in  σ ^ < σ < 1 ,  σ ^ > δ , which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces. [ABSTRACT FROM AUTHOR]

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of S[μn, +​]-polynomials and relate this concept to the known notion of number systems. The Riemann–Roch theorem for the ring Z of the integers that we obtained recently uses the understanding of Z as a ring of polynomials S[X] in one variable over the absolute base S, where 1+1=X+X². The absolute base S (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious F1​. [ABSTRACT FROM AUTHOR]

It is known that multiple zeta values whose weight and depth are of opposite parity can be written in terms of multiple zeta values of lower depth. This theorem is called parity result. Multiple zeta values are special values of the multiple polylogarithms and the parity result is generalized to functional relations satisfied by the multiple polylogarithms. In this paper, we consider q- and elliptic generalizations of the parity result. As a main result of this paper, we establish parity result for functions L k (a , α ; p , q) , which can be considered to be common deformations of q- and elliptic multiple polylogarithms. By taking the trigonometric and classical limits in the main theorem, we obtain q- and elliptic analogues of the parity result. [ABSTRACT FROM AUTHOR]

The first topic covered in [[1]] is built around the Gauss hypergeometric function HT ht , the Kummer hypergeometric function HT ht , and their generalization HT ht . Finally, paper [[11]] investigates certain identities associated with HT ht -binomial coefficients and HT ht -Stirling polynomials of the second kind. Hypergeometric Bernoulli numbers and polynomials connect to the hypergeometric function HT ht which is the subject of [[4]]. [Extracted from the article]

In this paper, a summation formula for a general family of a finite secant sum has been extended by making use of a particularly convenient integration contour method. The main theorem derived from this approach is the finite sum involving the Hurwitz-Lerch zeta function. This theorem for particular values is used to derive the finite product of the fifth roots of the quotient product of the gamma function along with finite sums and functional equations involving trigonometric functions. [ABSTRACT FROM AUTHOR]

In the paper, we prove that the set of discrete shifts of the Riemann zeta-function (ζ(s + 2πia1k), . . ., ζ(s + 2πiark)), k ∈ N, approximating analytic non-vanishing functions f1(s), . . ., fr(s) defined on {s ∈ C : 1/2 < Res < 1} has a positive density in the interval [N,N + M] with M = o(N), N → ∞, with real algebraic numbers a1, . . ., ar linearly independent over Q. A similar result is obtained for shifts of certain absolutely convergent Dirichlet series. [ABSTRACT FROM AUTHOR]

A very recent article delved into and expanded the four parametric linear Euler sums, revealing that two well-established subjects—Euler sums and series involving the zeta functions—display particular correlations. In this study, we present several closed forms of series involving zeta functions by using formulas for series associated with the zeta functions detailed in the aforementioned paper. Another closed form of series involving Riemann zeta functions is provided by utilizing a known identity for a series of rational functions in the series index, expressed in terms of Gamma functions. Furthermore, we demonstrate a myriad of applications and relationships of series involving the zeta functions and the extended parametric linear Euler sums. These include connections with Wallis's infinite product formula for π , Mathieu series, Mellin transforms, determinants of Laplacians, certain integrals expressed in terms of Euler sums, representations and evaluations of some integrals, and certain parametric Euler sum identities. The use of Mathematica for various approximation values and certain integral formulas is elaborated upon. Symmetry naturally occurs in Euler sums. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified Mellin transform of the product of the square of the Riemann zeta function and the exponentially decreasing function, and we discuss its probabilistic and approximation properties. It turns out that this Mellin transform approximates the identical zero in the strip { s ∈ C : 1 / 2 < σ < 1 } . [ABSTRACT FROM AUTHOR]

Recently a function was constructed that satisfies all known properties of a tree-level scattering of four massless scalars via the exchange of an infinite tower of particles with masses given by the non-trivial zeroes of the Riemann zeta function. A key ingredient in the construction is an even entire function whose only zeroes coincide with the non-trivial zeroes of the Riemann zeta function. In this paper we show that exactly the same conclusions can be drawn for an infinite class of even entire functions with only zeroes on the real line. This shows that the previous result does not seem to be connected to specific properties of the Riemann zeta function, but it applies more generally. As an application, we show that exactly the same conclusions can be drawn for L-functions other than the Riemann zeta function. [ABSTRACT FROM AUTHOR]

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In this paper, we introduce the U-Bernoulli, U-Euler, and U-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized U-Bernoulli, U-Euler and U-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the U-Bernoulli, U-Euler, and U-Genocchi polynomial matrices, which involves the generalized Pascal matrix. [ABSTRACT FROM AUTHOR]

In this paper, we study Multiple slit diffraction and n- array linear antennae in negative refractive index material. We show that the evanescent wave plays a vital role in the near-field term. The evanescent wave grows significantly, unlike in conventional materials, and satisfies a novel kind of convergence known as Cesàro convergence. We calculate the intensity of multiple slits and the antenna's amplification factor (AF) in terms of the Riemann zeta function. We further demonstrate that the Riemann zeta function gives rise to additional nulls. We deduce that all the diffraction scenarios in which the traveling wave satisfies the geometric series in the medium of the positive refractive index will enhance the evanescent wave, which satisfies Cesàro convergence in the medium of the negative refractive index. [ABSTRACT FROM AUTHOR]

In the paper, the approximation of analytic functions on compact sets of the strip { s = σ + i t ∈ C ∣ 1 / 2 < σ < 1 } by shifts F (ζ (s + i u 1 (τ)) , ... , ζ (s + i u r (τ))) , where ζ (s) is the Riemann zeta-function, u 1 , ... , u r are certain differentiable increasing functions, and F is a certain continuous operator in the space of analytic functions, is considered. It is obtained that the set of the above shifts in the interval [ T , T + H ] with H = o (T) , T → ∞ , has a positive lower density. Additionally, the positivity of a density with a certain exceptional condition is discussed. Examples of considered operators F are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the approximation of a certain class of analytic functions by shifts Z (s + i τ) , τ ∈ R , of the modified Mellin transform Z (s) of the square of the Riemann zeta-function ζ (1 / 2 + i t) . More precisely, we prove the existence of a closed non-empty set F such that there are infinitely many shifts Z (s + i τ) , which approximate a given analytic function from F with a given accuracy. In the proof, the weak convergence of measures in the space of analytic functions is applied. Then, the set F coincides with the support of a limit measure. [ABSTRACT FROM AUTHOR]

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case when G = SLn and P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding to n = 1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014. [ABSTRACT FROM AUTHOR]

A large family of relations among multiple zeta values may be described using the combinatorics of shuffle and quasi-shuffle algebras. While the structure of shuffle algebras has been well understood for some time now, quasi-shuffle algebras were only formally studied relatively recently. In particular, Hoffman gives a thorough discussion of the algebraic structure, including a choice of algebra basis, and applies his results to produce families of relations among multiple zeta values and their generalizations. In a recent paper, Hirose and Sato established a new family of relations coming from a new generalized shuffle structure, lifting a set of graded relations established by the author to genuine ungraded relations. In this paper, we define a commutative algebra structure on the space of noncommutative polynomials on a countable alphabet, generalizing the shuffle-like structure of Hirose and Sato. We show that, over the rational numbers, this generalized quasi-shuffle algebra is isomorphic to the standard shuffle algebra, allowing us to reproduce most of Hoffman's results on quasi-shuffle algebras. We then apply these results to the case of multiple zeta values, reproducing several known families of results and establishing several new families of linear relations. [ABSTRACT FROM AUTHOR]

In this paper we investigate the representation of integrals involving the product of the Legendre Chi function, polylogarithm function and log function. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed. [ABSTRACT FROM AUTHOR]

We prove an analogue of Kronecker's second limit formula for a continuous family of "indefinite zeta functions". Indefinite zeta functions were introduced in the author's previous paper as Mellin transforms of indefinite theta functions, as defined by Zwegers. Our formula is valid in dimension g = 2 at s = 1 or s = 0 . For a choice of parameters obeying a certain symmetry, an indefinite zeta function is a differenced ray class zeta function of a real quadratic field, and its special value at s = 0 was conjectured by Stark to be a logarithm of an algebraic unit. Our formula also permits practical high-precision computation of Stark ray class invariants. [ABSTRACT FROM AUTHOR]

In the paper, it is obtained that there are infinite discrete shifts Ξ (s + i k h) , h > 0 , k ∈ N 0 of the Mellin transform Ξ (s) of the square of the Riemann zeta-function, approximating a certain class of analytic functions. For the proof, a probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied. [ABSTRACT FROM AUTHOR]

Let a = { a m } and b = { b m } be two periodic sequences of complex numbers, and, additionally, a is multiplicative. In this paper, the joint approximation of a pair of analytic functions by shifts (ζ n T (s + i τ ; a) , ζ n T (s + i τ , α ; b)) of absolutely convergent Dirichlet series ζ n T (s ; a) and ζ n T (s , α ; b) involving the sequences a and b is considered. Here, n T → ∞ and n T ≪ T 2 as T → ∞ . The coefficients of these series tend to a m and b m , respectively. It is proved that the set of the above shifts in the interval [ 0 , T ] has a positive density. This generalizes and extends the Mishou joint universality theorem for the Riemann and Hurwitz zeta-functions. [ABSTRACT FROM AUTHOR]

In this paper, a theorem is obtained on the approximation in short intervals of a collection of analytic functions by shifts (ζ (s + i t k α 1) , ... , ζ (s + i t k α r)) of the Riemann zeta function. Here, { t k : k ∈ N } is the sequence of Gram numbers, and α 1 , ... , α r are different positive numbers not exceeding 1. It is proved that the above set of shifts in the interval [ N , N + M ] , here M = o (N) as N → ∞ , has a positive lower density. For the proof, a joint limit theorem in short intervals for weakly convergent probability measures is applied. [ABSTRACT FROM AUTHOR]

In 1973, Voronin proved the functional independence of the Riemann zeta-function ζ(s), i.e., that ζ(s) and its derivatives do not satisfy a certain equation with continuous functions. In the paper, we obtain a joint version of the Voronin theorem. Keywords: [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function Q.s/of the NcHO at s = n .n = 2; 3; : : : / and then the generating and meta-generating functions for Apéry-like numbers defined through the analysis of special values Q.n/. Actually, we show that the generating function w2n of such Apéry-like numbers appearing (as the "first anomaly") in Q.2n/for n = 2 gives an example of automorphic integral with rational period functions in the sense of Knopp, but still a better explanation remains to be clarified explicitly for n > 2. This is a generalization of our earlier result on showing that w2 is interpreted as a €.2/-modular form of weight 1. Moreover, certain congruence relations over primes for "normalized" Apérylike numbers are also proven. In order to describe w2n in a similar manner as w2, we introduce a differential Eisenstein series by using analytic continuation of a classical generalized Eisenstein series due to Berndt. The differential Eisenstein series is actually a typical example of the automorphic integral of negative weight. We then have an explicit expression of w4 in terms of the differential Eisenstein series. We discuss also shortly the Hecke operators acting on such automorphic integrals and relating Eichler's cohomology group. [ABSTRACT FROM AUTHOR]

In his lost notebook, Ramanujan noted down many elegant identities involving divisor functions and the modified K-Bessel function, and some of them are connected with the Fourier series expansion of the non-holomorphic Eisenstein series. Recently, Cohen established interesting generalizations of some of the identities of Ramanujan. In this paper, we study Ramanujan and Cohen-type identities associated to a generalized divisor function and the modified K-Bessel function. In the process, we extend a result of Chandrasekharan and Narasimhan and some identities of Cohen. Furthermore, we obtain a new identity for odd zeta values that can be thought of as a Bessel function analogue of Ramanujan's famous formula for odd zeta values. [ABSTRACT FROM AUTHOR]

An investigation into a family of definite integrals containing log-polylog functions with negative argument will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and some may be represented as a BBP type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function. [ABSTRACT FROM AUTHOR]

Raina and Srivastava [20] introduced a generalized Lambert transform. Goyal and Laddha [8] have introduced generalizations of the Riemann zeta function and generalized Lambert transform. In the present paper, we introduce generalizations of the Hurwitz-Lerch zeta function and Lambert transform in a diverse direction. We derive generating functions involving generalized Hurwitz-Lerch zeta function. Connections between the generalized Lambert transform and generalized Hurwitz-Lerch zeta function are established. An inversion formula for the generalized Lambert transform is obtained. Some examples and special cases to illustrate our results are also mentioned. [ABSTRACT FROM AUTHOR]

For any real β 0 ∈ [ 1 2 , 1) , let GRH [ β 0 ] be the assertion that for every Dirichlet character χ and all zeros ρ = β + i γ of L (s , χ) , one has β ⩽ β 0 (in particular, GRH [ 1 2 ] is the Generalized Riemann Hypothesis). In this paper, we show that the validity of GRH [ 9 10 ] depends only on certain distributional properties of the zeros of the Riemann zeta function ζ (s) . No conditions are imposed on the zeros of nonprincipal Dirichlet L-functions. [ABSTRACT FROM AUTHOR]

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series. [ABSTRACT FROM AUTHOR]

Suppose that Q is a positive defined n × n matrix, and Q [ x ̲ ] = x ̲ T Q x ̲ with x ̲ ∈ Z n . The Epstein zeta-function ζ (s ; Q) , s = σ + i t , is defined, for σ > n 2 , by the series ζ (s ; Q) = ∑ x ̲ ∈ Z n ∖ { 0 ̲ } (Q [ x ̲ ]) − s , and it has a meromorphic continuation to the whole complex plane. Let n ⩾ 4 be even, while φ (t) is an increasing differentiable function with a continuous monotonic bounded derivative φ ′ (t) such that φ (2 t) (φ ′ (t)) − 1 ≪ t , and the sequence { a φ (k) } is uniformly distributed modulo 1. In the paper, it is obtained that 1 N # N ⩽ k ⩽ 2 N : ζ (σ + i φ (k) ; Q) ∈ A , A ∈ B (C) , for σ > n − 1 2 , converges weakly to an explicitly given probability measure on (C , B (C)) as N → ∞ . [ABSTRACT FROM AUTHOR]