Generalized Bernoulli Numbers Research Articles

Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Canonical equivariant cohomology classes generating zeta values of totally real fields

It is known that the special values at nonpositive integers of a Dirichlet L 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 L -functions of totally real fields. Hecke L 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.

IDENTITIES INVOLVING GENERALIZED BERNOULLI NUMBERS AND PARTIAL BELL POLYNOMIALS WITH THEIR APPLICATIONS

In the present paper, we show that the partial Bell polynomials allow for obtaining identities involving the generalized Bernoulli numbers. Then, on applying these identities we derive different generating and bilateral generating functions.

Fekete polynomials, quadratic residues, and arithmetic

Fekete polynomials associate with each prime number p a polynomial with coefficients −1 or 1 except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo p. These polynomials were already considered in the 19th century in relation to the studies of Dirichlet L-functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomials. In particular, we propose two conjectures on the structure of these Galois groups. We also provide some computational evidence toward the validity of these conjectures.

Method for calculating Dirichlet L-functions

Recently, the authors gave asymptotic formulas for L(s, χ), which associated with a primitive Dirichlet character χ, in terms of the generalized Bernoulli numbers. In this paper, based on the aforementioned asymptotic formulas we describe a method for calculating Dirichlet L-functions that can be used to validate the generalized Riemann hypothesis up to a height T > 0. Our method is a refinement of the one presented by Davies and Haselgrove.

Analytic Continuations of Character and Alternating Tornheim Zeta Functions

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.

On p-rationality of [formula omitted] for Sophie Germain primes l

The p-rationality of a totally real abelian number field can be checked from the values L(2−p,χ) of Dirichlet L-functions, for all non-principal even Dirichlet characters associated to the field. Using this criterion and the properties of the generalized Bernoulli numbers, we study the p-rationality of Q(ζ2l+1)+, the maximal real subfield of Q(ζ2l+1), for Sophie Germain primes l and odd primes p that are primitive roots modulo l. We prove that Q(ζ2l+1)+ is p-rational for such pairs if p<4l. We also prove that the Siegel's heuristics on the equidistribution of the residues of Bernoulli numbers modulo p imply that Q(ζ2l+1)+ is p-rational for all but finitely many p that are primitive roots modulo l.

Complex numbers similar to the generalized Bernoulli numbers and their applications

Let χ be a primitive Dirichlet character modulo k ≥ 3. In this paper, we define complex numbers associated with χ, which we denote by Cr(χ) (r = 0, 1,…), and we discuss their properties and their relationships with the generalized Bernoulli numbers.

A twisted generalization of the classical Dedekind sum

In this paper, we express three different, yet related, character sums in terms of generalized Bernoulli numbers. Two of these sums are generalizations of sums introduced and studied by Berndt and Arakawa–Ibukiyama–Kaneko in the context of the theory of modular forms. A third sum generalizes a sum already studied by Ramanujan in the context of theta function identities. Our methods are elementary, relying only on basic facts from algebra and number theory.

Higher congruences between newforms and Eisenstein series of squarefree level

Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level $\Gamma_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert--Samuel multiplicities.

Cuspidal divisor class groups of non-split Cartan modular curves

I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal Divisor Class Group $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,\chi}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.

The extension theorem for the Lee and Euclidean weights over [formula omitted]

The Lee and Euclidean weights have the extension property over the local rings Z/pkZ, p prime. The non-vanishing of certain Fourier coefficients is established by expressing the coefficients in terms of generalized Bernoulli numbers and making use of knowledge of the locations of zeros of Dirichlet L-functions.

Reciprocal relations of Bernoulli and Euler numbers/polynomials

ABSTRACTBy means of the symmetric summation theorem on polynomial differences due to Chu and Magli [Summation formulae on reciprocal sequences. European J Combin. 2007;28(3):921–930], we examine Bernoulli and Euler polynomials of higher order. Several reciprocal relations on Bernoulli and Euler numbers and polynomials are established, including some recent ones obtained by Agoh Shortened recurrence relations for generalized Bernoulli numbers and polynomials. J Number Theory. 2017;176:149–173.

Generalized Bernoulli Numbers and Polynomials in the Context of the Clifford Analysis

Generalized Bernoulli Numbers and Polynomials in the Context of the Clifford Analysis

Une identité pour des polynômes d'Appell

In this paper, we establish an identity for some Appell polynomials generalizing explicit formulas for generalized Bernoulli numbers and polynomials.

Shortened recurrence relations for generalized Bernoulli numbers and polynomials

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.

Analysis of the p-adic q-Volkenborn integrals: An approach to generalized Apostol-type special numbers and polynomials and their applications

By applying the p-adic q-Volkenborn Integrals including the bosonic and the fermionic p-adic integrals on p-adic integers, we define generating functions, attached to the Dirichlet character, for the generalized Apostol-Bernoulli numbers and polynomials, the generalized Apostol-Euler numbers and polynomials, generalized Apostol-Daehee numbers and polynomials, and also generalized Apostol-Changhee numbers and polynomials. We investigate some properties of these numbers and polynomials with their generating functions. By using these generating functions and their functional equation, we give some identities and relations including the generalized Apostol-Daehee and Apostol-Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, Frobenious-Euler polynomials, the generalized Bernoulli numbers and the generalized Euler numbers and the Frobenious-Euler polynomials. By using the bosonic and the fermionic p-adic integrals, we derive integral represantations for the generalized Apostol-type Daehee numbers and the generalized Apostol-type Changhee numbers.

Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1

In this paper, two new series for the logarithm of the Γ-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to π−1. In particular, for any value of the form ln⁡Γ(12n±απ−1) and Ψk(12n±απ−1), where Ψk stands for the kth polygamma function, α is positive rational greater than 16π, n is integer and k is non-negative integer, these series have rational terms only. In the specified zones of convergence, derived series converge uniformly at the same rate as ∑(nlnm⁡n)−2, where m=1,2,3,… , depending on the order of the polygamma function. Explicit expansions into the series with rational coefficients are given for the most attracting values, such as ln⁡Γ(π−1), ln⁡Γ(2π−1), ln⁡Γ(12+π−1), Ψ(π−1), Ψ(12+π−1) and Ψk(π−1). Besides, in this article, the reader will also find a number of other series involving Stirling numbers, Gregory's coefficients (logarithmic numbers, also known as Bernoulli numbers of the second kind), Cauchy numbers and generalized Bernoulli numbers. Finally, several estimations and full asymptotics for Gregory's coefficients, for Cauchy numbers, for certain generalized Bernoulli numbers and for certain sums with the Stirling numbers are obtained. In particular, these include sharp bounds for Gregory's coefficients and for the Cauchy numbers of the second kind.

Functional Equations associated with Generalized Bernoulli Numbers and Polynomials

In this paper, we investigate the functional equations of the multiple Dirichlet and Hurwitz L-functions associated with Bernoulli numbers and polynomials attached to Dirichlet character.

Class Number Formula for Certain Imaginary Quadratic Fields

In this note we shall show how Carlitz in 1954 could have reached an analogue of the Voronoi congruence in the more difficult case of p≡1(mod4): h(-4p) ≡B(p+1)/2(x4)(mod p), where B(p+1)/2(x4) is the generalized Bernoulli number with x4 being the Kronecker symbol associated to the Gaussian field Q(√-4).