Yoshihiro Ônishi
In this paper we announce some results obtained for certain algebraic functions, which we call of cyclotomic type. The main results properly resemble von Staudt-Clausen's theorem and Kummer's congruence for the Bernoulli numbers, and such theorems for the Hurwitz numbers.
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Contour Integrations and Parity Results of Hurwitz-type Cyclotomic Euler Sums
本文通过轮廓积分和留数计算方法，研究了Hurwitz型cyclotomic欧拉和的奇偶性，并推导出相关公式，同时提出了关于多重Hurwitz多项式对数函数的两个猜想。
Hurwitz numbers are the Laurent coefficients of an elliptic function $\wp(u)$ of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher genus cases. They satisfy completely von Staudt-Clausen type theorem, an extension of von Staudt second theorem, and Kummer type congruence relation. The present paper is revised and combined version of math.NT/0304377 and math.NT/0312178 containing many numerical examples.
Symmetry Results for Cyclotomic Multiple Hurwitz Zeta Values via Contour Integrals
本文通过积分曲线方法推导出 cyclotomic 多变量 Hurwitz 范数值的显式对称公式，不依赖代数正则化，并应用于多重范数值的对称性研究。
Using three basic facts concerning Hurwitz zeta function,we give new natural proofs of the known results on Bernoulli polynomials,gamma function and also obtain Gauss' expression for Psi function at a rational point,all in a unified fashion.We also give a new proof of the relation between log gamma and derivative of the Hurwitz zeta function,including that of Stirling's expression for log gamma.
There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.
We prove nonlinear relation on multiple Hurwitz-Riemann zeta functions. Using analytic continuation of these multiple Hurwitz-Riemann zeta function, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers.
Recurrences for values of the Hurwitz type poly-Bernoulli numbers and polynomials
该研究针对一类新的广义Hurwitz型多重伯努利数与多项式，通过引入新的第二类Stirling数推广形式，推导其计算算法与负上指标组合公式，并建立该推广Stirling数与图论的联系。
On the Bernoulli--Hurwitz periods
本文将 Bernoulli–Hurwitz 测度周期刻画为带层数据 CM 曲线上的权一 Eisenstein 特殊值，联系 Katz 测度并推出普通椭圆曲线的周期公式与 $p$-进 Kronecker 第一极限公式。
Classically, Euler developed the theory of the Riemann zeta - function using as his starting point the exponential and partial fraction forms of cot(z) . In this paper we wish to develop the theory of $L$-functions of elliptic curves starting from the theory of elliptic functions in an analogous manner.
In this paper, we provide an alternative method to calculate the values of generalized multiple Hurwitz zeta function at non-positive integers by means of \emph{Raabe}'s formula and the \textit{Bernoulli} numbers.
The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are 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 number…
We give two proofs of a folkore result relating numerical semigroups of embedding dimension two and binary cyclotomic polynomials and explore some consequences. In particular, we give a more conceptual reproof of a result of Hong et al. (2012) on gaps between the exponents of non-zero monomials in a binary cyclotomic polynomial. The intent of the author with this expositional paper is to better unify the various results within the cyclotomic polynomial and numerical semigroup communities.
We give a result on the asymptotic behavior of the Hurwitz-Lerch multiple zeta functions near non-positive integer points by using the Apostol-Bernoulli polynomials. From this result, we can evaluate limit values at non-positive integer points.
We construct the q-extension of the Hurwitz's type L-function which interpolates the q-extension of generalized Bernoulli polynomials attached to $chi$.
In this paper, we generalize Bernoulli-Goss polynomials, and give a criterion on the divisibility of zeta functions of cyclotomic function fields. As an application of our criterion, for a given polynomial $f(u)$, we prove that there are infinitely many cyclotomic function fields whose zeta polynomial is divided by $f(u)$.
We give an instant evaluation of multiple Zeta function at non-positive integers by elementary methods and discuss the Fourier theory (on unit interval) of the product of Bernoulli polynomials.We also show that the polynomial expression for Hurwitz Zeta function(at non-positive integral values of the first variable) and the polynomial expression for Bernoulli polynomials are equivalent.
$p$-adic multiple $L$-functions and twisted multiple Bernoulli numbers
计算Furusho等人引入的$p$-adic多重$L$函数在正整数元组处的特殊值，通过给出循环多重伯努利数与扭曲多重伯努利数关系的显式公式，得到这些特殊值可表示为循环多重调和值的无穷和。
In this paper, we establish more identities of generalized multi poly-Euler polynomials with three parameters and obtain a kind of symmetrized generalization of the polynomials. Moreover, generalized multi poly-Bernoulli polynomials are defined using multiple polylogarithm and derive some properties parallel to those of poly-Bernoulli polynomials. These are generalized further using the concept of Hurwitz-Lerch multiple zeta values.
Contour Integration and Cyclotomic Apéry-Like Series Involving Generalized Binomial Coefficients
本文通过轮廓积分方法研究了一类涉及广义二项式系数的 cyclotomic 阿佩里型级数，推导出多重 polylogarithms 等表达式，并恢复了一些已知结果。
Cyclotomic norm congruences for zeta values and modular $L$-values
该研究在分圆塔中建立ζ函数与模L值的同余关系，在GL₁、GL₂情形下分别推导相关同余并给出应用，为相关数论问题提供了新的研究路径。
We find a representation for the Maclaurin coefficients of the Hurwitz zeta-function in terms of semi-convergent series involving the Bernoulli polynomials and the Stirling numbers of the first kind. In particular, this gives a representation for the coefficients of the Riemann zeta function. Our main instrument is a certain series transformation formula. A similar result is proved also for the Maclaurin coefficients of the Lerch zeta function.
In this paper, we introduce the concept of the (higher order) Appell-Carlitz numbers which unifies the definitions of several special numbers in positive characteristic, such as the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers.Their generating function is usually named Hurwitz series in the function field arithmetic. By using Hasse-Teichmüller derivatives, we also obtain several properties of the (higher order) Appell-Carlitz numbers, including a recurrence formula, two closed forms expressions, and a determinant expression. The recurrence formula implies Carlitz's recurrence formu…
In this paper, by introducing a new operation in the vector space of analytic functions, the author presents a method for derivating the well-known formulas: $ζ(1-k)=-\frac{B_k}{k}$ and $ζ(1-n,a)=-\frac{B_n(a)}{n}$ , where $ζ$, $ζ(1-n,a)$ denote the Riemann zeta function and the Hurwitz zeta function respectively. $B_k$ is the $k$-th Bernoulli number. Also the author steps further to deduce some identities related to Bernoulli number and Bernoulli polynomial. Moreover, when combining the operation with forward difference, we can show a new formula for Riemann zeta function, i.e. \[ζ(s)=e\sum_{…
We prove an Iwasawa Main Conjecture for the class group of the $\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field $\mathbb{F}_q(θ)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[θ]\,$), showing that its Fitting ideal is generated by a Stickelberger element. We use this and a link between the Stickelberger element and a $\mathfrak{p}$-adic $L$-function to prove a close analog of the Ferrero-Washington theorem for $\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations of the Bernoulli-Goss numbers $β(j)$ (i.e., on the values of the Goss $ζ$-function at…
A difference equation analogue of the Knizhnik-Zamolodchikov equation is exhibited by developing a theory of the generating function $H(z)$ of Hurwitz polyzeta functions to parallel that of the polylogarithms. By emulating the role of the KZ equation as a connection on a suitable bundle, a difference equation version of the notion of connection is developed for which $H(z)$ is a flat section. Solving a family of difference equations satisfied by the Hurwitz polyzetas leads to the normalized multiple Bernoulli polynomials (NMBPs) as the counterpart to the Hurwitz polyzeta functions, at tuples o…
We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (Nörlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $ζ(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $γ_1(a)$. The …
We present a summation rule using the Mellin transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials.
The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Its coefficients are the subject of intensive study and some formulas are known for them. Here we are interested in formulas which are valid for all natural numbers $n$. In these a host of famous number theoretical objects such as Bernoulli numbers, Stirling numbers of both kinds and Ramanujan sums make their appearance, sometimes even at the same time! In this paper we present a survey of these formulas which until now were scattered in the literature and introduce an unified approach…
For these two decades, the Arakawa-Kaneko zeta function has been studied actively. Recently Kaneko and Tsumura constructed its variants from the viewpoint of poly-Bernoulli numbers. In this paper, we generalize their zeta functions of Arakawa-Kaneko type to those with indices in which positive and negative integers are mixed. We show that values of these functions at positive integers can be expressed in terms of the multiple Hurwitz zeta star values.
On arithmetic nature of $q$-analogue of the generalized Stieltjes constants
本文拓展了2003年关于Hurwitz zeta函数$q$-模拟的研究，推导了其洛朗级数前两项系数的闭式，探究了相关超越性，提出了线性无关猜想并证明了超半数相关数在有理数域上线性无关。
We provide an explicit formula for the Tornheim double series in terms of integrals involving the Hurwitz zeta function. We also study the limit when the parameters of the Tornheim sum become natural numbers, and show that in that case it can be expressed in terms of definite integrals of triple products of Bernoulli polynomials and the Bernoulli function $A_k (q): = kζ'(1 - k,q)$.
By using q-Volkenborn integral on Z_{p}, we (simsek, simsekCanada) constructed new generating functions of the (h,q)-Bernoulli polynomials and numbers. By applying the Mellin transformation to the generating functions, we constructed integral representation of the twisted (h,q)-Hurwitz function and twisted (h,q)-two-variable L-function. By using these functions, we construct twisted new (h,q)-partial zeta function which interpolates the twisted (h,q)-Bernoulli polynomials and generalized twisted (h,q)-Bernoulli numbers at negative integers. We give relation between twisted (h,q)-partial zeta f…
$L$-function invariants for 3-manifolds and relations between generalized Bernoulli polynomials
该研究定义了负定型配管流形的$L$-函数，证明其为整函数且$s=0$处值对应Witten--Reshetikhin--Turaev不变量，还得到了这类推广型zeta函数负整数特殊值的线性关系。
Some reciprocity formulas for generalized Dedekind-Rademacher sums
本文通过建立Bernoulli函数乘积公式，结合多个解析公式，导出广义Dedekind-Rademacher和的互反公式，且多个经典互反定理可作为特例得出。
A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist--Meurman--type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group \cite{Tempesta1}-\cite{Tempesta3}. Their role in the theory of $L$--genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann--Hurwitz--type zeta functions.
We introduce some new higher dimensional generalizations of the Dedekind sums associated with the Bernoulli functions and of those Hardy sums which are defined by the sawtooth function. We generalize a variant of Parseval's formula for the discrete Fourier transform to derive finite trigonometric representations for these sums in a simple unified manner. We also consider a related sum involving the Hurwitz zeta function.
Ramanujan-type identities for alternating Hurwitz zeta functions
将拉马努金关于\(\zeta(2n + 1)\)的恒等式扩展到交错赫维茨zeta函数，研究其在不同模对称条件下性质及对应恒等式，还建立相关函数无穷级数表达式与变换公式。
The link between Vandiver's conjecture and Gauss sums is well known since the papers of Iwasawa (1975), Thaine (1995-1999) and Angl{è}s-Nuccio (2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver's conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the pth cyclotomic field (1984). Then we provide a specific use of Gauss sums of characters of order p of F\_ell^x and prove new criteria for Vandiver's conjecture to h…
The aim of this paper is to construct generating functions for some families of special finite sums with the aid of the Newton-Mercator series, hypergeometric series, and $p$-adic integral (the Volkenborn integral). By using these generating functions, their functional equations, and their partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials and others. Among…
We show that Riemann-Hurwitz-style translation formulas obtained by Kuz'min, Kida, Iwasawa, Wingberg et alii for the lambda invariant attached to certain Iwasawa moduli in cyclotomic Z{\ell}-extension of number fields are essentially equivalent. More precisely, we prove that all these formulas,including those stated in terms of representations, resul tidentically for purely algebraic reasons from the arithmetic computation of a suitable Herbrand quotient which it suffices to carry out in the cyclic case of prime degree {\ell}.
A famous conjecture bearing the name of Vandiver states that $p \nmid h_p^+$ in the $p$ - cyclotomic extension of $\Q$. Heuristics arguments of Washington, which have been briefly exposed in Lang (1978), p. 261 and Washington (1996), p. 158 suggest that the Vandiver conjecture should be false if certain conditions of statistical independence are fulfilled. In this note, we assume that Greenberg's conjecture is true for the \nth{p} cyclotomic extensions and prove an elementary consequence of the assumption that Vandiver's conjecture fails for a certain value of $p$: the result indicates that th…
Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups
本文证明分圆类群某些特征空间在相对密度为一的素数集合上消失或有界，支持 Vandiver 猜想与 Iwasawa 循环性猜想，并给出精确计算。
Recent work of Bettin and Conrey on the period functions of Eisenstein series naturally gave rise to the Dedekind-like sum \[ c_{a}\left(\frac{h}{k}\right) \ = \ k^{a}\sum_{m=1}^{k-1}\cot\left(\frac{πmh}{k}\right)ζ\left(-a,\frac{m}{k}\right), \] where $a\in\mathbb{C}$, $h$ and $k$ are positive coprime integers, and $ζ(a,x)$ denotes the Hurwitz zeta function. We derive a new reciprocity theorem for these Bettin--Conrey sums, which in the case of an odd negative integer $a$ can be explicitly given in terms of Bernoulli numbers. This, in turn, implies explicit formulas for the period functions ap…
The modified Bernoulli numbers $B_{n}^{*}$ considered by Zagier are generalized to modified Nörlund polynomials ${B_{n}^{(\ell)*}}$. For $\ell\in\mathbb{N}$, an explicit expression for the generating function for these polynomials is obtained. Evaluations of some spectacular integrals involving Chebyshev polynomials, and of a finite sum involving integrals of the Hurwitz zeta function are also obtained. New results about the $\ell$-fold convolution of the square hyperbolic secant distribution are obtained, such as a differential-difference equation satisfied by a logarithmic moment and a close…
An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv k$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers (mod $n$) in the set $\{1,2,\ldots,n\}$. This is an analogue of Euler's $ϕ$ function. We introduce the multidimensional generalization of $\varrho$, which is the analogue of Jordan's function. We establish identities for the power sums of regular integers (mod $n$) and for some other finite…
Linear independence of $q$-analogue of the generalized Stieltjes constants over number fields
本文将Chatterjee和Garg关于广义Stieltjes常数$q$-模拟的算术结果从有理数域推广到数域的两类情形，拓展了相关数论研究的范围。
We prove new exact formulas for the generalized sum-of-divisors functions, $σ_α(x) := \sum_{d|x} d^α$. The formulas for $σ_α(x)$ when $α\in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over all of the prime factors $n \leq x$ and terms involving the $r$-order harmonic number sequences and the Ramanujan sums $c_d(x)$. The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when $r > 1$ and are related to the generalized Bernoulli numbers when $r \leq 0$ is integer-valued. A key part of our new expansions of the Lambert series generating…
Let p be an odd prime. Let K = Q(zeta) be the p-cyclotomic field. Let v be any primitive root mod p. Let sigma be a Q-isomorphism of K. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 \in Z[G] where 1 \leq v^n \leq p-1 is a notation mod p. We apply a Kummer and Stickelberger relation of K to some singular not primary numbers A of K connected to p-class group C_p of K and prove they verify the congruence A^P(sigma) = 1 mod p^2. This p-adic method on singular numbers A allows us to prove: in a straightforward way the connection between relative p-class group C_p^- and the solutions o…
Appell Sequences in Analytic Number Theory: Shifted Expansions and Dirichlet L-Functions
该研究将阿佩尔移位定理置于博雷尔-拉普拉斯框架，扩展阿佩尔-梅林框架至狄利克雷L-函数，构造生成函数并确立多项式R_n^χ的五项结构性质，相关结果为首次报道。
Three notes on Ser's and Hasse's representations for the zeta-functions
本文研究zeta函数的Ser与Hasse级数表示，证明二者等价，推导了涉及多种特殊数与多项式的zeta函数、Hurwitz zeta函数及Dirichlet级数的新级数表示，还得到Stieltjes常数等的新展开式。
The Kaleidoscopic Filter on the Partition Manifold, Part I: The Simplicial Ehrhart, Barycentric Pascal Product, and Continuous Bernoulli Exact Closed Forms for $p(n)$
本文通过将划分多面体嵌入有理多面体理论，推导出整数划分函数p_k(n)的精确O(1)公式，并扩展至无限制划分函数p(n)。
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