Finite and symmetric Mordell-Tornheim multiple zeta values

Mordell-Tornheim多重ζ函数、其积分类比及多重Polylogarithms之间的关系

Mordell-Tornheim multiple zeta-functions, their integral analogues, and relations among multiple polylogarithms

本文研究了Mordell-Tornheim多重ζ函数及其积分类比在x=0处的渐进行为，并通过比较两种渐近公式建立了多重Polylogarithms之间的非平凡关系。

Mordell-Tornheim zeta 函数：Kronecker 极限型公式、级数求值及其应用

The Mordell-Tornheim zeta function: Kronecker limit type formula, Series Evaluations and Applications

本文为 Mordell-Tornheim zeta 函数建立 Kronecker 极限型公式，统一多种经典模关系，并由此得到级数求值与新的混合函数方程族。

Mordell--Tornheim zeta function：Kronecker 极限型公式与特殊值

Mordell--Tornheim zeta function: Kronecker limit type formulas and Special values

本文为广义 Mordell--Tornheim zeta 函数建立了 Kronecker 极限型公式与级数求值，导出混合泛函方程，并研究了变量相等时的零点、特殊值与奇点。

Mordell--Tornheim Zeta 值、其交错版本及其有限模拟

Mordell--Tornheim Zeta Values, Their Alternating Version, and Their Finite Analogs

本文研究经典及交错 Mordell--Tornheim zeta 值生成空间，否证相关猜想，并研究有限和的超同余，推广至有限模拟，提出新猜想。

On functional relations for the alternating analogues of Tornheim's double zeta function

We give new proofs of two functional relations for the alternating analogues of Tornheim's double zeta function. Using the functional relations, we give new proofs of some evaluation formulas found by H. Tsumura for these alternating series.

A functional relation for Tornheim's double zeta functions

In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results for the behavior of a certain Witten's zeta function at each integer. As an appendix, we show a functional equation for Euler's double ze…

Simple proof of the functional relation for the Lerch type Tornheim double zeta function

In this paper, we give a simple proof of the functional relation for the Lerch type Tornheim double zeta function. By using it, we obtain simple proofs of some explicit evaluation formulas for double $L$-values.

Functional Equations of Tornheim Zeta Function and the alpha-calculus of r-th order derivative of Zeta(s,alpha)

For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the tool,we obtain with ease an expression for T(s,s,s) for any complex s;and evaluate T(n,n,n) for any integer n>=1;and also evaluate T(n1,0,n2)for integers n1,n2>1 with n1+n2 odd.We obtain for Tornheim zeta function,the counterparts of Euler's formula for zeta(2n)and its analogue for zeta(2n+1),where Zeta(s) is Riemann zeta function.

Tornheim-like series, harmonic numbers and zeta values

Explicit evaluations of the Tornheim-like double series in the form \[ \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in \mathbb{N\cup } \left\{ 0 \right\} \] and their extensions are given. Furthermore, series of the type \[ \sum_{m=1}^\infty \frac{2H_{2m+1}-H_{m}}{2m\left( 2m+1 \right)} \] and some other Tornheim-like multiple series are evaluated in terms of the zeta values.

Rapidly convergent series representations of symmetric Tornheim double zeta functions

In the present paper, for $s,t,u \in {\mathbb{C}}$, we show rapidly (or globally) convergent series representations of the Tornheim double zeta function $T(s,t,u)$ and (desingularized) symmetric Tornheim double zeta functions. As a corollary, we give a new a proof of known results on the values of $T(s,s,s)$ at non-positive integers and the location of the poles of $T(s,s,s)$. Furthermore, we prove that the function $T(s,s,s)$ can not be written by a polynomial in the form of $\sum_{k=1}^j c_k \prod_{r=1}^q ζ^{d_{kr}} (a_{kr} s + b_{kr})$, where $a_{kr}, b_{kr}, c_k \in {\mathbb{C}}$ and $d_{k…

Identities for Bernoulli polynomials related to multiple Tornheim zeta functions

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear factors. The special case of Bernoulli numbers has important applications in the study of multiple Tornheim zeta functions. The proof of the main result relies on properties of Eulerian polynomials and higher-order Bernoulli polynomials.

Evaluation of Tornheim's type of double series

We examine values of certain Tornheim's type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_2$, proposed by Komori, Matsumoto and Tsumura, is given.

Mordell-Tornheim ζ函数与Herglotz-Zagier型函数的函数方程

Mordell-Tornheim zeta functions and functional equations for Herglotz-Zagier type functions

本文通过推广Mordell-Tornheim ζ函数特例$Θ(z,x)$，建立其与Herglotz-Zagier函数的联系，导出长期 sought 的$Φ_k(x)$函数方程，并借助Gauss和、Fekete多项式给出无穷多Herglotz型积分方程。

Apostol-Vu双重ζ函数的中值定理及其应用

Mean value theorems for the Apostol-Vu double zeta-function and its application

该研究推导得出Apostol-Vu型及Mordell-Tornheim型双重ζ函数的均方值，为相关数论研究提供了中值定理层面的结果支撑。

The evaluation of Tornheim double sums. Part2

We provide an explicit formula for the Tornheim double series T(a,0,c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a=m, c=n, we show that in the most interesting case of even weight N:=m+n the Tornheim sum T(m,0,n) can be expressed in terms of zeta values and the family of integrals % \int_0^1 loggamma(q) B_{k}(q) Cl_{j+1} (2 πq) dq, % with k+j = N, where B_{k}(q) is a Bernoulli polynomial and \Cl_{j+1}(x) is a Clausen function.

On the parity result for multiple Dirichlet series

In this paper, we discuss the parity result for multiple Dirichlet series which contains some special values of multiple zeta functions as special cases, Mordell--Tornheim type of multiple zeta values, zeta values of the root systems and so on. Moreover, we can give explicit expression in terms of lower series by using main theorem.

On an analog of the Arakawa-Kaneko zeta function and relations of some multiple zeta values

T. Ito defined an analog of the Arakawa-Kaneko zeta function to obtain relations among Mordell-Tornheim multiple zeta values. In this paper, we develop two things related to an analog of the Arakawa-Kaneko zeta function. One is to find an analog of the Arakawa-Kaneko zeta function of Miyagawa-type (defined by T. Miyagawa) and to obtain a relation among Miyagawa-type MZVs. The other is to find a class of zeta functions to which Ito's zeta functions of the case of general index are related.

A Note on Colored Tornheim's Double Series

In this short note, we provide an explicit formula to compute every colored double Tornheim's series by using double polylogarithm values at roots of unity. When the colors are given by $\pm 1$ our formula is different from that of Tsumura [On alternating analogues of Tornheim's double series II, Ramanujan J. 18 (2009), 81-90] even though numerical data confirm both are correct in almost all the cases. This agreement can also be checked rigorously by using regularized double shuffle relations of the alternating double zeta values in weights less than eight.

Finite multiple zeta values associated with 2-colored rooted trees

We define finite multiple zeta values (FMZVs) associated with some combinatorial objects, which we call 2-colored rooted trees, and prove that FMZVs associated with 2-colored rooted trees satisfying certain mild assumptions can be written explicitly as $\mathbb{Z}$-linear combinations of the usual FMZVs. Our result can be regarded as a generalization of Kamano's recent work on finite Mordell-Tornheim multiple zeta values. As an application, we will give a new proof of the shuffle relation of FMZVs, which was first proved by Kaneko and Zagier.

Mean values of the Barnes double zeta-function

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_{…

Truncated $t$-adic symmetric multiple zeta values and double shuffle relations

We study a refinement of the symmetric multiple zeta value, called the $t$-adic symmetric multiple zeta value, by considering its finite truncation. More precisely, two kinds of regularizations (harmonic and shuffle) give two kinds of the $t$-adic symmetric multiple zeta values, thus we introduce two kinds of truncations correspondingly. Then we show that our truncations tend to the corresponding $t$-adic symmetric multiple zeta values, and satisfy the harmonic and shuffle relations, respectively. This gives a new proof of the double shuffle relations for $t$-adic symmetric multiple zeta value…

Desingularization of multiple zeta-functions of generalized Hurwitz-Lerch type and evaluation of p-adic multiple L-functions at arbitrary integers

We study analytic properties of multiple zeta-functions of generalized Hurwitz-Lerch type. First, as a special type of them, we consider multiple zeta-functions of generalized Euler-Zagier-Lerch type and investigate their analytic properties which were already announced in our previous paper. Next we give `desingularization' of multiple zeta-functions of generalized Hurwitz-Lerch type, which include those of generalized Euler-Zagier-Lerch type, the Mordell-Tornheim type, and so on. As a result, the desingularized multiple zeta-function turns out to be an entire function and can be expressed as…

欧拉双重zeta值的$q$-模拟之间的代数恒等式

Algebraic identities among $q$- analogue of Euler double zeta values

本文探究另一类q-双重zeta函数变体及其星型变体的恒等式，借助q>1的Nielsen反射公式q-模拟，研究相关恒等式，分析参数趋于0的行为并与经典情形对比，还讨论Mordell-Tornheim r重zeta函数q-模拟及其与q-双重zeta函数的关系。

On algebra of big zeta values

The algebra of big zeta values we introduce in this paper is an intermediate object between multiple zeta values and periods of the multiple zeta motive. It consists of number series generalizing multiple zeta values, the simplest examples, which are not multiple zeta series, are Tornheim sums. We show that convergent big zeta values are periods of the moduli space of stable curves of genus zero on one hand and multiple zeta values on the other hand. It gives an alternative way to prove that any such period may be expressed as a rational linear combination of multiple zeta values and a simple …

多重zeta值到有根森林的推广

Generalisations of multiple zeta values to rooted forests

该研究证明收敛的洗牌树化zeta值有级数表示，提出多重zeta值到有根森林的新推广并研究其代数性质，推导相关经典结果的初等证明，刻画锥zeta值与树化zeta值的关系并给出显式求值公式。

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned a…