Taekyun Kim
Recently (see [1]) I has introduced an interesting the Euler-Barnes multiple zeta function. In this paper we construct the q-analogue of Euler-Barnes multiple zeta function which interpolates the q-analogue of Frobenius-Euler numbers of higher order at negative integers.
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We construct Barnes' type Changhee q-zeta function.
The Barnes multiple zeta function is useful to study in the number theory and Knot thoey and Mathematical Physics. In this paper we consider q-extension of Barnes type multiple zeta function and we also construct the q-extension of Euler polynomials of higher order
In this paper, we introduce $q$-analogues of the Barnes multiple zeta functions. We show that these functions can be extended meromorphically to the whole plane, and moreover, tend to the Barnes multiple zeta functions when $q\uparrow 1$ for all complex numbers.
On the mean values of the Barnes multiple zeta function
研究Barnes多重ζ函数均值的渐进行为，通过分析其与黎曼ζ函数的关系，揭示其渐近特性。
In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan's classical formula for the Riemann zeta values can be derived from functional equations for Barnes zeta-functions. In the latter half part, we generalize some evaluation formulas of certain series involving hyperbolic functions in terms of Bernoulli polynomials. The original formulas were classically given by Cauchy, Mellin, Ramanujan, and later recovered and formulated by Berndt. From our consideration, we give multiple ve…
The \emph{Barnes $ζ$-function} is \[ ζ_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} \] defined for $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to $\C$. Specialized at negative integers $-k$, the Barnes $ζ$-function gives \[ ζ_n (-k, x; \a) = \frac{(-1)^n k!}{(k+n)!} \, B_{k+n} (x; \a) \] where $B_k(x; \a)$ is a \emph{Bernoulli--Barnes polynomial}, which can be also defined through a generating function that has a slightly more general form than that for Bernoulli polynomials. Specializing $B_k(0; \a)$ gives the \emph{Bernoul…
In the study of order estimation of the Riemann zeta-function $ ζ(s) = \sum_{n=1}^\infty n^{-s} $, solving Lindelöf hypothesis is an important theme. As one of the relationships, asymptotic behavior of mean values has been studied. Furthermore, the theory of the mean values is also noted in the double zeta-functions, and the mean values of the Euler-Zagier type of double zeta-function and Mordell-Tornheim type of double zeta-function were studied. In this paper, we prove asymptotic formulas for mean square values of the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{…
As one of the asymptotic formulas of the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In this paper, we prove an approximate functional equation of the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (α+vm+wn)^{-s} $. Also, applying this approximate functional equation and the van der Corput method, we obtain upper bounds for $ ζ_2(1/2 + it, α; v, w) $ and $ ζ_2(3/2 + it, α; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.
Let $a, w_1, w_2,\cdot\cdot\cdot, w_r >0$ and $s \in \mathbb{C}$. We put $w= (w_1,\cdot\cdot\cdot,w_r)$. Then the Barnes $r$-ple zeta function is defined by $ζ_r(s, w, a) = \sum_{m_1=0}^{\infty} \cdot\cdot\cdot \sum_{m_r=0}^{\infty} 1/(a+m_1w_1+\cdot\cdot\cdot +m_rw_r)^s$ when $σ:= \Re(s)>r$. In this paper, we show that the Barnes double zeta function $ζ_2(σ, w, a)$ has real zeros in the interval $(1,2)$ if and only if $0< a < (w_1+w_2)/2$ and the number of such zero is precisely one if $0< a< (w_1+w_2)/2$.
Mean values and upper bounds for the Hurwitz and Barnes multiple zeta functions
本文研究Hurwitz型和Barnes型多重ζ函数的均值平方值，通过线性组合经典Hurwitzζ函数的方法，建立参数σ的渐进行为和上界，揭示Barnes型参数向量的算术结构对均值平方值的影响。
Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s ζ_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $ζ_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-p…
Functional equation, upper bounds and analogue of Lindelöf hypothesis for the Barnes double zeta function
本文研究Barnes双ζ函数的函数方程，并利用Phragmén-Lindelöf凸性原理，推导出该函数在t趋于无穷时的上界。
We construct the q-analogue of Euler-Barnes' numbers and polynomials, and investigate their some properties.
Suppose that $ω_1,\ldots,ω_N$ are positive real numbers and $x$ is a complex number with positive real part. The multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ with parameter vector $\barω=(ω_1,\ldots,ω_N)$ is defined as a deformation of the Barnes multiple zeta function as follows $$ ζ_{E,N}(s,x;\barω)=\sum_{t_1=0}^\infty\cdots\sum_{t_N=0}^\infty \frac{(-1)^{t_1+\cdots+t_N}}{(x+ω_1t_1+\cdots+ω_Nt_N)^s}. $$ In this paper, based on the fermionic $p$-adic integral, we define the $p$-adic analogue of multiple Barnes-Euler zeta function $ζ_{E,N}(s,x;\barω)$ which we denote by $ζ_{p,E,N}(…
Analogues of the Lindelöf Hypothesis for the Barnes multiple zeta function and related problems
研究Barnes多重zeta函数的Lindelöf假设类比，通过积分均值建立等价条件，并揭示依赖于有理秩的独特现象。
We first survey the known results on functional equations for the double zeta-function of Euler type and its various generalizations. Then we prove two new functional equations for double series of Euler-Hurwitz-Barnes type with complex coefficients. The first one is of general nature, while the second one is valid when the coefficients are Fourier coefficients of a cusp form.
Laurent series expansions for the Barnes multiple zeta function
本文研究Barnes多重zeta函数的Laurent系数，在最高极点给出显式极限公式，在较低极点建立与Taylor系数的全阶关系，并利用整函数性质分析大阶行为，推广了常数项的有限部分表示。
By using $q$-Volkenborn integration and uniform differentiable on $\mathbb{Z}%_{p}$, we construct $p$-adic $q$-zeta functions. These functions interpolate the $q$-Bernoulli numbers and polynomials. The value of $p$-adic $q$-zeta functions at negative integers are given explicitly. We also define new generating functions of $q$-Bernoulli numbers and polynomials. By using these functions, we prove analytic continuation of some basic (or $q$-) $L$% -series. These generating functions also interpolate Barnes' type Changhee $% q $-Bernoulli numbers with attached to Dirichlet character as well. By a…
We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta function. Especially, we prove that the bilateral zeta function has a nice Fourier series expansion and the Barnes zeta function can be expressed as a finite sum of bilateral zeta functions. By these properties of the bilateral zeta functions, We obtain simple proofs of some formulas, for example the reflection formula for the multiple gamma function, the inversio…
Integral Representations of Three Novel Multiple Zeta Functions for Barnes Type: A Probabilistic Approach
本文通过超几何概率分布研究三种新型Barnes型多重ζ函数的积分表示，发现其中两个函数可解析延拓为整个复平面上的整函数，拓展了传统Barnes多重ζ函数的应用范围。
Shintani-Barnes cocycles and values of the zeta functions of algebraic number fields
本文通过引入指数扰动新技术，推广 Eisenstein 上闭链构造，建立了 Shintani-Barnes 上闭链，从而统一给出一般数域 zeta 函数在正整数处的值。
On the Laurent series expansions of the Barnes double zeta function
本文研究Barnes双ζ函数在s=1和s=2处的Laurent级数展开，推导出系数的极限表达式，得到类似于欧拉-斯蒂尔杰斯常数的类比，并分析其渐进行为，揭示其与经典理论的区别。
The variation of Barnes and Bessel zeta functions
针对几何中含圆锥奇点的物理与分析问题中的Barnes和Bessel泽塔函数，通过复分析方法计算其零点导数及关于参数的变分，得到整数参数下Barnes泽塔函数变分的新表达式与Bessel泽塔函数变分的两种表达式并验证等价性。
Using the Jackson integral, we obtain the $q$-integral analogue of the Raabe type formulas for Barnes multiple Hurwitz-Lerch zeta functions and Barnes and Vardi's multiple gamma functions. Our results generalize $q$-integral analogue of the Raabe type formulas for the Hurwitz zeta functions and log gamma functions in [N. Kurokawa, K. Mimachi, and M. Wakayama, Jackson's integral of the Hurwitz zeta function, Rend. Circ. Mat. Palermo (2) 56 (2007), no. 1, 43--56]. During the proof we also obtain a new formula on the relationship between the higher and lower orders Hurwitz zeta functions.
We give explicit expressions (or at least an algorithm of obtaining such expressions) of the coefficients of the Laurent series expansions of the Euler-Zagier multiple zeta-functions at any integer points. The main tools are the Mellin-Barnes integral formula and the harmonic product formulas. The Mellin-Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence.
Mixed Berndt-Type Integrals and Generalized Barnes Multiple Zeta Functions
本文提出混合伯恩特型积分及其闭式公式，并通过广义巴克斯多重ζ函数进行评估。
The classical Lerch's formula states the following normalized product: $$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2π}}{Γ(x)},\quad \textrm{Re}(x)>0, $$ where $Γ(x)$ is the Euler gamma function. In this note, by using Barnes' multiple zeta function and its alternating form, we obtain two kinds of generalizations of Lerch's formula, which imply the product $$ \prod_{n=1}^\infty n=\sqrt{2π} $$ (in the sense of zeta regularization) and the product $$\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot 5}\frac{6\cdot6}{5\cdot 7}\cdots=\fracπ{2}$$ (Wallis' formula in 1656), respectively.
We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes Gamma function, including the particular case also known as multiple Gamma function.
The aim of this paper is to construct new Dedekind type sums. We construct generating functions of Barnes' type multiple Frobenius-Euler numbers and polynomials. By applying Mellin transformation to these functions, we define Barnes' type multiple l-functions, which interpolate Frobenius-Euler numbers at negative integers. By using generalizations of the Frobenius-Euler functions, we define generalized Dedekind type sums and prove corresponding reciprocity law. We also give twisted versions of the Frobenius-Euler polynomials and new Dedekind type sums and corresponding reciprocity law. Further…
We study the behavior of partially twisted multiple zeta-functions. We give new closed and explicit formulas for special values at non-positive integer points of such zeta-functions. Our method is based on a result of M. de Crisenoy on the fully twisted case and the Mellin-Barnes integral formula.
The Barnes-Hurwitz zeta cocycle at $s=0$ and Ehrhart quasi-polynomials of triangles
本文基于Hayes定理，对PGL₂(ℚ)上某1-上循环在s=0处的特殊值给出几何解释，并由此得到上循环显式公式、Hayes定理的推广与新证明，以及ℝ²中三角形Ehrhart拟多项式零次系数的求和公式。
We introduce an "$L$-function" $\mathcal{L}$ built up from the integral representation of the Barnes' multiple zeta function $ζ$. Unlike the latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action of a group $G$. Although these two functions differ from each other, we can use $\mathcal{L}$ to study $ζ$. In fact, the transformation formula for $\mathcal{L}$ under $G$-transformations provides us with a new perspective on the special values of both $ζ$ and its $s$-derivative. In particular, we obtain Kronecker limit formulas for $ζ$ when restricted to points fixed by eleme…
A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions
本文研究了一类伯恩德特积分的计算方法，通过留数定理和级数展开，将其转化为更简单的求和形式，并用Gamma函数和√π表达结果，同时关联到巴纳茨多重ζ函数。
Recently, the higher-order q-Euler polynomials and multiple q-Euler zeta functions are introduced by T. Kim ([8, 9]). In this paper, we investigate some symmetric properties of the multiple q-Euler zeta function and derive various identities concerning the higher-order q-Euler polynomials from the symmetric properties of the multiple q-Euler zeta functions.
Recently, T. Kim considered Euler zeta function which interpolates Euler polynomials at negative integer (see [3]). In this paper, we study degenerate Euler zeta function which is holomorphic function on complex s-plane associated with degenerate Euler polynomials at negative integers.
In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.
The purpose this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta function which interpolates multiple q-Euler numbers at negative integers.
The fact that the double zeta values at n and m can be written as a sum of products of two zeta values and of zeta value at m+n, whenever n+m is odd is due to Euler. We shall show a weak version of this result for the Galois l-adic realization.
In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.
The distribution of the zeros of the Euler double zeta-function $ζ_2(s_1,s_2)$, in the case when $s_1=s_2$, is studied numerically. Some similarity to the distribution of the zeros of Hurwitz zeta-functions is observed.
The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler's results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. In this note, we establish a q-analog of Euler's decomposition formula. More specifically, we show that Euler's decomposition formula can be extended to what might be referred to as a ``double q-zeta function'' in such a way that Euler's formula is recovered in the limit as q tends to 1.
In this paper, we consstruct a new extended q-Bernoulli numbers and poly nomials. From these numbers, we derive the multiple zeta functions and give some relations between multiple Bernoulli numbers and multiple zeta functions.
We give new closed and explicit formulas for "multiple zeta values" at non-positive integers of generalized Euler-Zagier multiple zeta-functions. We first prove these formulas for a small convenient class of these multiple zeta-functions and then use the analyticity of the values on the parameters defining the multiple zeta-functions to deduce the formulas in the general case. Also, for our aim we prove an extension of "Raabe's lemma" due to E. Friedman and A. Pereira.
We present several formulas for some specific multiple $L$-values of conductor four. This grew out from the study of zeta functions of level four of Arakawa-Kaneko type. Closely related is a new version of multiple poly-Euler numbers and we briefly discuss this too.
In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as …
We prove that the Shintani zeta function associated to the space of binary cubic forms cannot be written as a finite sum of Euler products. Our proof also extends to several closely related Dirichlet series. This answers a question of Wright in the negative.
The alternating multiple harmonic sums are partial sums of the infinite series defining the Euler sums which are the alternating version of the multiple zeta value series. In this paper, we present some systematic structural results of the van Hamme type congruences of these sums, collected as finite Euler sums. Moreover, we relate this to the structure of the Euler sums, which generalizes the corresponding result of the multiple zeta values. We also provide a few conjectures with extensive numerical support.
We consider partial zeta functions $ζ(s,c)$ associated with ray classes $c$'s of a totally real field. Stark's conjecture implies that an appropriate product of $\exp(ζ'(0,c))$'s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp(ζ'(0,c))$ in terms of Barnes' multiple gamma function. Yoshida ``decomposed'' Shintani's formula: he defined the symbol $X(c,ι)$ satisfying that $\exp(ζ'(0,c))=\prod_ι \exp(X(c,ι))$ where $ι$ runs over all real embeddings of $F$. Hence we can decompose a Stark unit into a product of $[F:\mathbb Q]$ terms. The main result i…
Multiple zeta values and Euler's reflection formula for the gamma function
本文利用Hoffman调和代数与二项式恒等式，为源自Gamma函数Euler反射公式的恒等式提供了纯代数证明。
Explicit formulas of the relation between multiple zeta functions of Arakawa-Kaneko and Euler-Zagier types
本文给出Arakawa-Kaneko型多重zeta函数表示为Euler-Zagier型多重zeta函数与多重zeta值乘积的Q-线性组合的显式公式，并作为证明关键显式建立multi-polylogarithm函数间的函数方程。
Explicit Evaluation of Euler-Apéry Type Multiple Zeta Star Values and Multiple $t$-Star Values
本文通过迭代积分建立递推关系，进而给出Euler-Apéry型多重zeta星值及其参数变体和多重$t$-星值的显式求值。
Generalized Euler decomposition formula for interpolated multiple zeta values
本文提出了一种通用的t-shuffle乘积公式，用于推导插值多重ζ值的广义欧拉分解公式，并通过两种方法验证了高度为一的情况。
We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $ζ(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.
For Hurwitz zeta function, we obtain power series expression in second variable for its higher order derivatives (with respect to first variable) at non-positive integer arguments and consequently obtain rapidly decreasing series expression for Riemann zeta fuction at positive odd integer arguments. Further, we obtain corresponding results for Dirichlet L-series. We also a unified proof of various classical identities involving Riemann zeta values.
On a Class of Berndt-type Integrals and Related Barnes Multiple Zeta Functions
本文利用围道积分、留数定理及 Jacobi 椭圆函数的级数技术，将仅含双曲余弦的 Berndt 型积分化为规范 Ramanujan 型级数，得到 Γ(1/4) 与 π^{-1/2} 的有理多项式显式表达式，并关联 Barnes 多重 zeta 函数。
We study the analytic behaviour of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato-Tate conjectures we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.
In 1776, L. Euler proposed three methods, called prima methodus, secunda methodus and tertia methodus, to calculate formulae for double zeta values. However strictly speaking, his last two methods are mathematically incomplete and require more precise reformulation and more sophisticated arguments for their justification. In this paper, we reformulate his formulae, give their rigorous proofs and also clarify that the formulae can be derived from the extended double shuffle relations.
The absolute zeta function for a scheme $X$ of finite type over $\mathbb{Z}$ satisfying a certain condition is defined as the limit as $p\to 1$ of the congruent zeta function for $X\otimes\mathbb{F}_p$. In 2016, after calculating absolute zeta functions for a few specific schemes, Kurokawa suggested that an absolute zeta function for a general scheme of finite type over $\mathbb{Z}$ should have an infinite product structure which he called the absolute Euler product. In this article, formulating his suggestion using a torsion free Noetherian $\mathbb{F}_1$-scheme defined by Connes and Consani,…
We provide a $q$-analogue of Euler's formula for $ζ(2k)$ for $k\in\mathbb{Z}^+$. Our main results are stated in Theorems 3.1 and 3.2 below. The result generalizes a recent result of Z.W. Sun who obtained $q$-analogues of $ζ(2)=π^2/6$ and $ζ(4)=π^4/90$.
We show that for a non-positive value of the first variable,Hurwitz zeta function becomes a polynomial in the second variable. We show this, using 'integration approach', instead of 'power series approach', which we had resorted to, in our earlier paper with the same title. This, in particular, explains why Riemann zeta function at positive even integer arguments, can be evaluated and why it cannot be evaluated explicitly at positive odd integer arguments.
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