Ilham A. Aliev, Ayhan Dil
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we construct certain analogues of the Arakawa-Kaneko zeta functions. We prove functional relations between these functions and the Mordell-Tornheim multiple zeta functions. Furthermore we give some formulas among Mordell-Tornheim multiple zeta values as their applications.
We introduce finite and symmetric Mordell-Tornheim type of multiple zeta values and give a new approach to the Kaneko-Zagier conjecture stating that the finite and symmetric multiple zeta values satisfy the same relations.
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.
We introduce the unified double zeta function of Mordell--Tornheim type and compute its values at non-positive integer points. We then discuss a possible generalization of the Kaneko--Zagier conjecture for all integer points.
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…
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.
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.
We prove that any Mordell-Tornheim sum with positive integer arguments can be expressed as a rational linear combination of multiple zeta values of the same weight and depth. By a result of Tsumura, it follows that any Mordell-Tornheim sum with weight and depth of opposite parity can be expressed as a rational linear combination of products of multiple zeta values of lower depth.
In this paper, we shall show that certain signed cyclic sums of Mordell-Tornheim L-values are rational linear combinations of products of multiple L-values of lower depths (i.e., reducible). This simultaneously generalizes some results of Subbarao and Sitaramachandrarao, and Matsumoto et al. As a direct corollary, we can prove that for any integer k>2 and positive integer n, the Mordell-Tornheim sums zeta_\MT(\{n\}_k) is reducible.
Analytic properties of three types of multiple zeta functions, that is, the Euler-Zagier type, the Mordell-Tornheim type and the Apostol-Vu type have been studied by a lot of authors. In particular, in the study of multiple zeta functions of the Apostol-Vu type, a generalized multiple zeta function, including both the Euler-Zagier type and the Apostol-Vu type, was introduced.In this paper, similarly we consider generalized multiple zeta-functions and $ L $-functions, which include both the Euler-Zagier type and the Mordell-Tornheim type as special cases.We prove the meromorphic continuation to…
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…
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。
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 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 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。
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.
Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a linear combination of Tornheim double series of the same weight. New closed form evaluations of various Euler sums are presented. Finally certain combinations of linear Euler sums that are reducible to Riemann zeta values are discovered.
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.
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.
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。
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.
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.
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.
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.
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_{…
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。
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…
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 …
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。
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。
In this paper, we settle the problem posed by Singer which is on a comparison problem between the renormalized values of shuffle type and of harmonic type of multiple zeta functions.
$p$-adic multiple zeta values and binomial multiple harmonic sums
本文提出了一种方法，用于推导$p$-adic多重ζ值的显式公式，引入了二项多重调和和作为关键工具。
Some identities on degenerate harmonic and degenerate higher-order harmonic numbers
本文研究退化调和数与退化高阶调和数，证明其相关无穷和可用退化zeta函数及退化Hurwitz zeta函数表示。
We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.
Joint extreme values of the Riemann zeta function at harmonic points
本文通过共振方法改进了黎曼zi函数在调和点处联合极值的估计，提供了新的行为见解，并恢复了部分已知极值结果。
We introduce adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values. They are two variants of cyclotomic multiple zeta values, closely related to each other. They arise as key tools for the study of $p$-adic cyclotomic multiple zeta values. Moreover, cyclotomic multiple harmonic values provide an adelic lift to a cyclotomic generalization of finite multiple zeta values. We establish certain standard properties of these two objects. We consider two types of properties : some related to double shuffle relations, and some related to associator and Kashiwara-Vergne relatio…
In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical and power series.
We calculate the special values of the spectral zeta function of the non-commutative harmonic oscillator, and give a general formula for them as integrals of certain algebraic functions. This is a generalization of the result by Ichinose-Wakayama (2005), in which the first two special values are studied.
We define and apply a method to study the non-vanishing of $p$-adic cyclotomic multiple zeta values. We prove the non-vanishing of certain cyclotomic multiple harmonic sums, and, via a formula proved in another paper, which expresses a cyclotomic multiple harmonic sums as an infinite sum of products of $p$-adic cyclotomic multiple zeta values, this implies the non-vanishing of certain $p$-adic cyclotomic multiple zeta values.
We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently conjectured by J. Choi, and give several more families of identities of a similar nature.
A variation of multiple $L$-values, which arises from the description of the special values of the spectral zeta function of the non-commutative harmonic oscillator, is introduced. In some special cases, we show that its generating function can be written in terms of the gamma functions. This result enables us to obtain explicit evaluations of them.
Following the Mellin and inverse Mellin transform techniques presented in our paper arXiv:1606.02150 (NT), we have established close forms of Laurent series expansions of products of bi- and trigamma functions /psi(z)*/psi(-z) and /psi_(1)(z)*/psi_(1)(-z). These series were used to find summation rules which include generalized harmonic numbers of first, second and third powers and values of the Riemann zeta-functions at integers / Bernoulli numbers, for example 2*Sum_(k-1)^(infinity)(H_(k)^((2))/k^3)=6*/zeta(2)*/zeta(3)-9*/zeta(5). Some of these rules were tested numerically.
The Kaneko-Zagier conjecture states that finite and symmetric multiple zeta values satisfy the same relations. In the previous work with H.~Bachmann and Y.~Takeyama, we proved that the finite and symmetric multiple zeta value are obtained as an `algebraic' and `analytic' limit at $q\rightarrow 1$ of a certain truncated multiple harmonic $q$-series, and studied its relations in order to give partial evidence of the Kaneko-Zagier conjecture. In this paper, we start with truncated multiple harmonic $q$-series of level $N$, which is a $q$-analogue of the truncated colored multiple zeta value. We i…
$p$-adic cyclotomic multiple zeta values depend on the choice of a number of iterations of the crystalline Frobenius of the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,μ_{N},\infty\}$. In this paper we study how the iterated Frobenius depends on the number of iterations, in relation with the computation of $p$-adic cyclotomic multiple zeta values in terms of cyclotomic multiple harmonic sums. This provides new results on that computation and the definition of a new pro-unipotent harmonic action.
This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$μ_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,μ_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$μ_{N}$'s involve sequences of positive integers ; in this IV-1, by consid…
Pro-unipotent harmonic actions and a computation of $p$-adic cyclotomic multiple zeta values
本文引入 pro-unipotent 调和作用，建立 $p$-进分圆多重 zeta 值与分圆多重调和和的互推公式，证明并推广相关猜想，同时给出有限分圆多重 zeta 值空间的维数界。
Recently, the present authors jointly with Tauraso found a family of binomial identities for multiple harmonic sums (MHS) on strings $(\{2\}^a,c,\{2\}^b)$ that appeared to be useful for proving new congruences for MHS as well as new relations for multiple zeta values. Very recently, Zhao generalized this set of MHS identities to strings with repetitions of the above patterns and, as an application, proved the two-one formula for multiple zeta star values conjectured by Ohno and Zudilin. In this paper, we extend our approach to $q$-binomial identities and prove $q$-analogues of two-one formulas…
In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.
A generalization of the Apery-like numbers, which is used to describe the special values $ζ_Q(2)$ and $ζ_Q(3)$ of the spectral zeta function for the non-commutative harmonic oscillator, are introduced and studied. In fact, we give a recurrence relation for them, which shows a ladder structure among them. Further, we consider the `rational part' of the higher Apery-like numbers. We discuss several kinds of congruence relations among them, which are regarded as an analogue of the ones among Apery numbers.
In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to certain rank one Drinfeld modules over Tate algebras in positive characteristic by Anglès, Tavares Ribeiro and the first author. We prove, by using techniques introduced by the second author, that various infinite families of such sums may be interpolated by polynomials, and we deduce, among several other results, properties of analogues of finite zeta values b…
Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized harmonic numbers carries information about the values of the arguments of Hurwitz function. In particular we prove: $\forall k\in \mathbb{N}:$ $ζ(k,1)\allowbreak =\allowbreak \frac{2^{k-1}}{2^{k-1}-1}% \sum_{n=1}^{\infty }\frac{H_{n}^{(k-1)}}{n2^{n}},$ where $H_{n}^{(k)}$ are defined below generalized harmonic numbers. Further we find generating function of th…
In this paper we present many new families of identities for multiple harmonic sums using binomial coefficients. Some of these generalize a few recent results of Hessami Pilehrood et al. As applications we prove several conjectures involving multiple zeta star values (MZSV): the Two-one formula conjectured by Ohno and Zudilin, and a few conjectures of Imatomi et al. involving 2-3-2-1 type MZSV, where 2 means some finite string of 2's
Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline pro-unipotent fundamental groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta values via explicit formulas ; this is in particular a study of the harmoni…
Multiple zeta values are real numbers defined by an infinite series generalizing values of the Riemann zeta function at positive integers. Finite truncations of this series are called multiple harmonic sums and are known to have interesting arithmetic properties. When the truncation point is one less than a prime $p$, the mod $p$ values of multiple harmonic sums are called finite multiple zeta values. The present work introduces a new class of congruence for multiple harmonic sums, which we call weighted congruences. These congruences can hold modulo arbitrarily large powers of $p$. Unlike res…
Higher-order degenerate harmonic numbers related to degenerate Riemann zeta function
本文研究了与退化Riemann zeta函数相关的高阶退化调和数、高阶退化超调和数及高阶交替退化超调和数，推导了它们的生成函数、显式表达式及相互关系。
We derive an expression for the value $ζ_Q(3)$ of the spectral zeta function $ζ_Q(s)$ studied by Ichinose and Wakayama for the non-commutative harmonic oscillator defined in the work of Parmeggiani and Wakayama using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted $J_2(n)$ and $J_3(n)$, which can be regarded as analogues of the Apéry numbers, naturally arise and play a key role in obtaining the expressions for the values $ζ_Q(2)$ and $ζ_Q(3)$. We also show that the numbers $J_2(n)$ and $J_3(n)$ have congruence relations like those of the Apéry numb…
P. Flajolet and B. Salvy \cite{FS1998} prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are reducible to zeta values, quadratic and linear sums. Moreover, we pro…
We consider the sum $\sum 1/γ$, where $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0,T]$, and consider the behaviour of the sum as $T \to\infty$. We show that, after subtracting a smooth approximation $\frac{1}{4π} \log^2(T/2π),$ the sum tends to a limit $H \approx -0.0171594$ which can be expressed as an integral. We calculate $H$ to high accuracy, using a method which has error $O((\log T)/T^2)$. Our results improve on earlier results by Hassani and other authors.
In recent years, there has been intensive research on the ${\mathbb Q}$-linear relations between multiple zeta (star) values. In this paper, we prove many families of identities involving the $q$-analog of these values, from which we can always recover the corresponding classical identities by taking $q\to 1$. The main result of the paper is the duality relations between multiple zeta star values and Euler sums and their $q$-analogs, which are generalizations of the Two-one formula and some multiple harmonic sum identities and their $q$-analogs proved by the authors recently. Such duality rela…
In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta function. Moreover, we provide explicit evaluations for almost all Euler sums with weight 5, which can be expressed in terms of zeta values and polylogarithms. Furthermore, we give explicit formula for several classes of Euler-related sums in terms of zeta values and harmonic numbers, and several examples are given. The given representations are new.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。