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The zeta function is like a milestone in analytic number theory. Considering Euler's definition of the zeta function in this article, we will find some recursion at even integral points. We will also discuss the Dirichlet eta function, even Bernoulli numbers, and the gamma–zeta relation. Toward the end, we will define something known as the odd zeta function and find a recurrence for the same. [ABSTRACT FROM AUTHOR]

We denote by $ \Delta _\nu $ Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues $ \beta _m, \ m\in \mathbb {Z}_+ $ β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $ \Delta _\nu $ Δ ν . Using a suitable polynomial decomposition of the multiplicity of each $ \beta _m $ β m , we write down a trace formula for the heat operator associated with $ \Delta _\nu $ Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $ t\searrow 0^+ $ t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $ \Delta _\nu $ Δ ν . [ABSTRACT FROM AUTHOR]

We study a sequence of constants known as the Bendersky-Adamchik constants which appear quite naturally in number theory and generalize the classical Glaisher-Kinkelin constant. Our main initial purpose is to elucidate the close relation between the logarithm of these constants and the Ramanujan summation of certain divergent series. In addition, we also present a remarkable, and previously unknown, expansion of the logarithm of these constants in convergent series involving the Bernoulli numbers of the second kind. [ABSTRACT FROM AUTHOR]

In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

We present a remarkably simple and surprisingly natural interpretation of the values of zeta functions at negative integers and zero. Namely, we are able to relate these values to areas related to partial sums of powers. We apply these results to further interpretations of values of L-functions at negative integers. We hint in a very brief way at some expected connections of this work with other current efforts to understand the mysteries of the values of zeta functions at integers. [ABSTRACT FROM AUTHOR]

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]

Several trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established. [ABSTRACT FROM AUTHOR]

It is known that the special values at nonpositive integers of a Dirichlet L-function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L-functions of totally real fields. Hecke L-functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393–417], we newly construct a canonical class, which we call the Shintani generating class , in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shintani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions. [ABSTRACT FROM AUTHOR]

The celebrated Wallis sequence Wn, which is defined by ..., is known to have the limit π/2 as n → ∞. Without using the Bernoulli numbers Bn, the authors present several asymptotic expansions and a recurrence relation for determining the coefficients of each asymptotic expansion related to the Wallis sequence Wn and the newly-introduced constants D and E, which are analogous to the Glaisher-Kinkelin constant A and the Choi-Srivastava constants B and C. [ABSTRACT FROM AUTHOR]

The "Problem section" of the School Science & Mathematics journal invites readers to share mathematical problems and solutions. The section features various proposed problems from contributors worldwide, including Romania, France, India, and more. Readers are encouraged to submit solutions to the problems before the specified deadline. [Extracted from the article]

Rademacher symbols may be defined in terms of Dedekind sums, and give the value at zero of the zeta function associated to a narrow ideal class of a real quadratic field. Duke extended these symbols to give the zeta function values at all negative integers. Here we prove Duke's conjecture that these higher Rademacher symbols are integer valued, making the above zeta value denominators as simple as the corresponding Riemann zeta value denominators. The proof uses detailed properties of Bernoulli numbers, including a generalization of the Kummer congruences. [ABSTRACT FROM AUTHOR]

The orbifold Euler characteristic of the moduli space ℳ g ; 1 of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: χ (ℳ g ; 1) = ζ (1 − 2 g) = − B 2 g / (2 g) , where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics. [ABSTRACT FROM AUTHOR]

In this paper, we apply the combinatorial results on counting permutations with fixed pinnacle and vale sets to evaluate the special values of the spectral zeta functions of Sturm–Liouville differential operators. As applications, we get a combinatorial formula for the special values of spectral zeta functions and give a new explicit formula for Bernoulli numbers. [ABSTRACT FROM AUTHOR]

In this paper, we use two different approaches to introduce q-analogs of Riemann's zeta function and prove that their values at even integers are related to the q-Bernoulli and q-Euler's numbers introduced by Ismail and Mansour [Analysis and Applications, 17(6) (2019), 853–895]. [ABSTRACT FROM AUTHOR]

We define a two-variable Dirichlet series associated with two arithmetic functions, which is related to the Riemann zeta function, the Dirichlet L-function, the Dirichlet series associated to the harmonic numbers, and truncated multiple zeta functions. Using the periodic Euler-Maclaurin summation formula, we obtain a representation in terms of an ordinary Dirichlet series, which leads to the explicit evaluation of its values at nonpositive integers. We also find a reciprocity formula, which provides some symmetric formulas involving Bernoulli and associated numbers. [ABSTRACT FROM AUTHOR]

We consider a multi-variable generalization of the zeta function defined by Arakawa and Kaneko. We establish its analytic continuation to an entire function, whence it follows that values at non-positive integers are expressed as multi-indexed poly-Bernoulli numbers of C-type. We also prove that its values at positive integers are in the space spanned by multiple zeta values. [ABSTRACT FROM AUTHOR]

A method developed by Crandall is used to analytically continue character and alternating analogues of a double sum in three complex variables, known as the Tornheim or Mordell–Tornheim–Witten zeta function. We evaluate these functions at triples of nonpositive integer arguments in terms of generalized Bernoulli numbers, and determine those for which a given function is zero. Some special cases are related to tangent numbers. [ABSTRACT FROM AUTHOR]

We study rather general multiple zeta functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta functions at non-positive integer points. We first treat the case when the polynomials are power sums, and observe that some "trivial zeros" exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those special values or period integrals. We also mention certain relations among Bernoulli numbers which can be deduced from our explicit formulas. Our proof of explicit formulas are based on the Euler–Maclaurin summation formula, Mahler's theorem, and a Raabe-type lemma due to Friedman and Pereira. [ABSTRACT FROM AUTHOR]

The Hurwitz zeta function ζ (s , a) admits a well-known (divergent) asymptotic expansion in powers of 1 / a involving the Bernoulli numbers. Using Wilson orthogonal polynomials, we determine an effective bound for the error made when this asymptotic series is replaced by nearly diagonal Padé approximants. By specialization, we obtain new fast converging sequences of rational approximations to the values of the Riemann zeta function at every integers ≥2. The latter can be viewed, in a certain sense, as analogues of Apéry's celebrated sequences of rational approximations to ζ (2) and ζ (3). [ABSTRACT FROM AUTHOR]

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function η(s), and hence Riemann's function ζ(s), is obtained in terms of the Exponential Integral function Es(iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions ζ±(s) and η±(s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints. [ABSTRACT FROM AUTHOR]

Four classes of multiple series, which can be considered as multifold counterparts of four classical summation formulae related to the Riemann zeta series, are evaluated in closed form. [ABSTRACT FROM AUTHOR]

We study the asymptotic expansion of the determinants of the graph Laplacians associated to discretizations of a half-translation surface endowed with a unitary flat vector bundle. By doing so, over the discretizations, we relate the asymptotic expansion of the number of spanning trees and the partition function of cycle-rooted spanning forests, weighted by the monodromy of the unitary connection of the vector bundle, to the corresponding zeta-regularized determinants. [ABSTRACT FROM AUTHOR]

We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2n) = ηnπ2n, we obtain the generating functions of the sequences ηn and (−1)nηn. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2n) and ζ(2n + 1). Furthermore, we prove some series equations for ∑ k = 1 ∞ − 1 k − 1 ζ p + k / k. [ABSTRACT FROM AUTHOR]

The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given. [ABSTRACT FROM AUTHOR]

Abstract In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two non-zero neighbouring Bernoulli numbers and analyses the approximating accuracy of the double inequality. [ABSTRACT FROM AUTHOR]

It is known that the special values of multiple zeta functions at non-positive arguments are indeterminate in most cases due to the occurrences of infinitely many singularities. In order to give a suitable rigorous meaning of the special values there, Furusho, Komori, Matsumoto, and Tsumura introduced the desingularized values by the desingularization method to resolve all singularities. While, Ebrahimi-Fard, Manchon, and Singer introduced the renormalized values to keep the "shuffle" relation by the renormalization procedure à la Connes and Kreimer. In this article, we reveal an equivalence, that is, an explicit interrelationship between these two values. As a corollary, we also obtain an explicit formula to describe renormalized values in terms of Bernoulli numbers. [ABSTRACT FROM AUTHOR]

In this paper, by applying a certain Saalschütz-Gelfand type formula for generalized Bernoulli numbers, we first study shortened recurrence relations for the values of the Dirichlet L-function attached to a primitive character at nonpositive integers and then derive some detreminantal expressions for these values. Further, we discuss several kinds of infinite sums involving the Riemann zeta values at negative odd integers and establish some closed formulas for them by making use of special identities related to the reciprocals of central binomial numbers. [ABSTRACT FROM AUTHOR]

In this paper we consider a new type of the q-Apostol Bernoulli numbers and polynomials. Firstly, we define the q-Apostol Bernoulli numbers and polynomials by making use of their generating function. Also, we observe many properties, i.e., the recurrence formula, the difference equation, the differential relation. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers (cf. [40,43,45,75]). The dimensions are expressed by special values of Shintani zeta functions for spaces of symmetric matrices at non-positive integers. This formula was given by Shintani for only a small part of the geometric side of the trace formula (see [64]). To be precise, it is the contribution of unipotent elements corresponding to the partitions (2 j , 1 2 n − 2 j) , where n denotes the degree and 0 ≤ j ≤ n. Hence, our work is to show that all the other contributions vanish. Therefore, one finds that Shintani's formula means the dimension itself. Combining our formula and an explicit formula of the Shintani zeta functions, which was discovered by Ibukiyama and Saito (cf. [44,42,61]), we can derive an explicit dimension formula for the principal congruence subgroups of level greater than two (cf. [43,45]). In this explicit dimension formula, the dimensions are described by degree n , weight k , level N , and the Bernoulli numbers B m. [ABSTRACT FROM AUTHOR]

In 2006, Kimoto and Wakayama discussed one kind of Apéry-like numbers which occurs in a representation of the special value of the spectral zeta function, and proposed a supercongruence conjecture on the sum of these numbers. This supercongruence conjecture was first proved by Long, Osburn and Swisher. In this paper, we extend the result of Long, Osburn and Swisher to a supercongruence modulo p 4 , which was originally conjectured by Sun. [ABSTRACT FROM AUTHOR]

Using the restricted and associated Stirling numbers of the second kind, we define the incomplete multi-poly-Bernoulli numbers which generalize poly-Bernoulli numbers. We study their analytic and combinatorial properties. As an application, we present a new infinite series representation of the multiple zeta values via the Lambert W-function. We also give similar results for incomplete Bernoulli numbers of Hurwitz type and incomplete multi-poly-Bernoulli-star numbers. [ABSTRACT FROM AUTHOR]

Using Parseval’s identity for the Fourier coefficients of x k , we provide a new proof that ζ ( 2 k ) = ( − 1 ) k + 1 B 2 k ( 2 π ) 2 k 2 ( 2 k ) ! . [ABSTRACT FROM AUTHOR]

In this paper, we are going to evaluate a family { E p ( 2 m , n , k ) | p ∈ Z } of sums of multiple zeta values with even arguments and polynomial weights defined as E p ( 2 m , n , k ) = ∑ α 1 + ⋯ + α k = n ∏ j = 1 k ( α j + p − 1 α j ) × ζ ( 2 m α 1 , 2 m α 2 , ⋯ , 2 m α k ) , where p is a nonnegative integer and m , n , k are positive integers with 1 ≤ k ≤ n . The special case E 0 ( 2 , n , k ) = ∑ α 1 + ⋯ + α k = n ζ ( 2 α 1 , 2 α 2 , ⋯ , 2 α k ) for positive integers α 1 , α 2 , ⋯ , α r with α r ≥ 2 was evaluated by Hoffman [7] in 2017 as ( − 1 ) n − k − 1 π 2 n ( 2 n + 1 ) ! ∑ i = 0 n − k ( n − i k ) ( 2 n + 1 2 i ) ( 2 2 i − 2 ) B 2 i , where B n ( n = 0 , 1 , 2 , ⋯ ) are Bernoulli numbers. [ABSTRACT FROM AUTHOR]

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers H ˜ n occur in the Laurent expansion about the origin of a certain Weierstrass ℘ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general ℘ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the ℘ function. [ABSTRACT FROM AUTHOR]

Wang and Cai established the following harmonic congruence for odd prime and positive integer : where denote the set of positive integers which are prime to . In this paper, the authors generalize it into super congruence for odd prime and positive integer : where and are positive integers such that [ABSTRACT FROM AUTHOR]

Extending Eulerian polynomials and Faulhaber's formula, 1 we study several combinatorial aspects of harmonic sums and polylogarithms at non-positive multi-indices as well as their structure. Our techniques are based on the combinatorics of noncommutative generating series in the shuffle Hopf algebras giving a global process to renormalize the divergent polyzetas at non-positive multi-indices. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Two combinatorial identities are proved. The first one allows easy to establish connections between Bernoulli and Stirling numbers and values of zeta function of integer arguments and to find numerous identities, including the new ones, for these quantities. It also leads to a simple derivation of Abel's analog of the Newton binomial formula, to the shortest proof of Staudt-Clausen theorem on Bernoulli numbers, and to a convenient summation formula for powers of natural numbers and series including such powers. The second identity presents the sum of two prime powers xp + yp (p = 2n + 1 = 6m ± 1) of real or natural numbers as Chebyshev polynomial of 4xy=(x + y)². The same sum is expressed as a sum of m subsequent powers of [xy(x + y)]²(xy + x² + y²)3 with coefficients (n - j - 1|2j)/(2j + 1). [ABSTRACT FROM AUTHOR]

We evaluate each Stieltjes constant уm as a finite sum involving the first m+ 1 Bernoulli numbers Bκ and the first m+1 derivatives (-1)κακ of the alternating zeta function at the point 1. In turn, we compute each ακ in an efficient way by means of a series with geometric rate ⅓. The coefficients of such a series are bounded and slowly decrease to zero. The computational significance of the preceding results is also discussed. [ABSTRACT FROM AUTHOR]

It is the main purpose of this paper to study shortened recurrence relations for generalized Bernoulli numbers and polynomials attached to χ , χ being a primitive Dirichlet character, in which some of the preceding numbers or polynomials are completely excluded. As a result, we are able to establish several kinds of such type recurrences by generalizing some known identities on classical Bernoulli numbers and polynomials such as Saalschütz–Gelfand and von Ettingshausen–Stern's formulas. Furthermore, we discuss shortened recurrence relations for special values of the Riemann zeta and its allied functions. [ABSTRACT FROM AUTHOR]

We define Cauchy numbers and polynomials associated with hypergeometric Bernoulli numbers, and study their properties. Moreover, we consider zeta functions that interpolate these Cauchy polynomials. Some special values of the zeta functions are expressed in terms of finite zeta star values or the Bernoulli numbers of higher order. [ABSTRACT FROM AUTHOR]

In this paper we investigate some interesting symmetric identities for twisted q-Euler polynomials of higher order in complex field. [ABSTRACT FROM AUTHOR]

For , let be the sum of all multiple zeta values with even arguments whose weight is and whose depth is . Of course is the value of the Riemann zeta function at , and it is well known that . Recently Shen and Cai gave formulas for and in terms of and . We give two formulas for , both valid for arbitrary , one of which generalizes the Shen-Cai results; by comparing the two we obtain a Bernoulli-number identity. We also give explicit generating functions for the numbers and for the analogous numbers defined using multiple zeta-star values of even arguments. [ABSTRACT FROM AUTHOR]

In the paper, we prove that the Bateman's G--function satisfies the double inequality Σ2m n=1 (2n-1)B2n/nx2n < G(x) - 1/x < Σ2m-1 n=1 (2n-1)B2n/nx2n , m ϵ N with best bounds, where B'rs are the Bernoulli numbers and we study the monotonicity of some functions involving the function G(x). Also, we present some estimates for the error term of a class of the alternating series, which improve and generalize some recent resutls and we prove the increasing monotonicity of a sequence arising from computation of the intersecting probability between a plane couple and a convex body. [ABSTRACT FROM AUTHOR]

We introduce the method of desingularization of multi-variable multiple zeta-functions (of the generalized Euler-Zagier type), under the motivation of finding a suitable rigorous meaning of the values of multiple zeta-functions at non-positive integer points. We reveal that multiple zeta-functions (which are known to be meromorphic in the whole space with infinitely many singular hyperplanes) turn out to be entire on the whole space after taking the desingularization. The desingularized function is given by a suitable finite "linear" combination of multiple zeta-functions with some arguments shifted. It is shown that specific combinations of Bernoulli numbers attain the special values at their non-positive integers of the desingularized ones.We also discuss twisted multiple zeta-functions, which can be continued to entire functions, and their special values at non-positive integer points can be explicitly calculated. [ABSTRACT FROM AUTHOR]

We construct p-adic multiple L-functions in several variables, which are generalizations of the classical Kubota-Leopoldt p-adic L-functions, by using a specific p-adic measure. Our construction is from the p-adic analytic side of view, and we establish various fundamental properties of these functions. (a) Evaluation at non-positive integers: We establish their intimate connection with the complex multiple zeta-functions by showing that the special values of the p-adic multiple L-functions at non-positive integers are expressed by the twisted multiple Bernoulli numbers, which are the special values of the complex multiple zeta-functions at non-positive integers. (b) Multiple Kummer congruences: We extend Kummer congruences for Bernoulli numbers to congruences for the twisted multiple Bernoulli numbers. (c) Functional relations with a parity condition: We extend the vanishing property of the Kubota-Leopoldt p-adic L-functions with odd characters to our p-adic multiple L-functions. (d) Evaluation at positive integers: We establish their close relationship with the p-adic twisted multiple star polylogarithms by showing that the special values of the p-adic multiple L-functions at positive integers are described by those of the p-adic twisted multiple star polylogarithms at roots of unity, which generalizes the result of Coleman in the single variable case. [ABSTRACT FROM AUTHOR]

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 ) . [ABSTRACT FROM AUTHOR]

In this paper, we give a proof of a conjecture made by Zagier in [4] about the inverse of some matrix related to double zeta values of parity ( even , odd ) . As a result, we obtain a family of Bernoulli number identities. We further generalize this family to a more general setting involving binomial coefficients of negative arguments. [ABSTRACT FROM AUTHOR]