Hideki Murahara, Takuya Murakami
We prove a new linear relation for multiple zeta values. This is a natural generalization of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.
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We prove a new linear relation for a q-analogue of multiple zeta values. It is a q-extension of the restricted sum formula obtained by Eie, Liaw and Ong for multiple zeta values.
Two classes of relations for multiple zeta values are handled algebraically. A restricted sum formula is proved by Eie, Liaw and Ong. The derivation relation is proved by Ihara, Kaneko and Zagier. In this paper we show the latter implies the former.
We give some restricted sum formulas for double zeta values whose arguments satisfy certain congruence conditions modulo 2 or 6, and also give an application to identities showed by Ramanujan for sums of products of Bernoulli numbers with a gap of 6.
Let Q(4n,d) be the sum of all multiple zeta values of depth d and weight 4n whose arguments are all multiples of 4. In this paper we derive a formula of Q(4n,d) for all d>2 as a finite sum involving binomial coefficients, Bernoulli numbers and the quantities Q(4m,2). The number of terms is about 3d^2/8 which is independent of n.
Using the combinatorial description of shuffle product, we prove or reformulate several shuffle product formulas of multiple zeta values, including a general formula of the shuffle product of two multiple zeta values, some restricted shuffle product formulas of the product of two multiple zeta values, and a restricted shuffle product formula of the product of $n$ multiple zeta values.
The Igusa Zeta function of restricted power series over $\mathbb{Q}_p$
本文研究 $\mathbb{Q}_p$ 上受限幂级数的 Igusa zeta 函数是否由低次项决定，发现单变量反例，但在关于 Newton 多面体充分非退化时成立。
We study multiple zeta values (MZVs) from the viewpoint of zeta-functions associated with the root systems which we have studied in our previous papers. In fact, the $r$-ple zeta-functions of Euler-Zagier type can be regarded as the zeta-function associated with a certain sub-root system of type $C_r$. Hence, by the action of the Weyl group, we can find new aspects of MZVs which imply that the well-known formula for MZVs given by Hoffman and Zagier coincides with Witten's volume formula associated with the above sub-root system of type $C_r$. Also, from this observation, we can prove some new …
In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively.
The cyclic sum formulas for multiple zeta and zeta-star values were respectively proved by Hoffman and Ohno, and Ohno and Wakabayashi. Kawasaki and Oyama obtained an analogous formulas for finite multiple zeta and zeta-star values. In this paper, we give a generalization of Kawasaki and Oyama's results.
The even weight period polynomial relations in the double shuffle Lie algebra $\mathfrak{ds}$ were discovered by Ihara, and completely classified by the second author by relating them to restricted even period polynomials associated to cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. In an article published in the same year, Gangl, Kaneko and Zagier displayed certain linear combinations of odd-component double zeta values which are equal to scalar multiples of simple zeta values in even weight, and also related them to restricted even period polynomials. In this paper, we relate the two sets of rela…
For odd $k$, we give a formula for the relations between double zeta values $ζ(r,k-r)$ with $r$ even. This formula provides a connection with the space of cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. This is the odd weight analogue of a result by Gangl, Kaneko and Zagier. We also provide an answer of a question asked by Zagier about the left kernel of some matrix. Although the restricted sum statement fails in the odd weight case, we provide an asymptotical statement that replaces it. Our statement works more generally for restricted sums with any congruence condition on the first entry of the d…
The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.
The sum formula for finite and symmetric multiple zeta values, established by Wakabayashi and the authors, implies that if the weight and depth are fixed and the specified component is required to be more than one, then the values sum up to a rational multiple of the analogue of the Riemann zeta value. We prove that the result remains true if we further demand that the component should be more than two or that another component should also be more than one.
We introduce an algebraic formulation of cyclic sum formulas for multiple zeta values and for multiple zeta-star values. We also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta-star values by reducing them to Kawashima relation.
We obtain a weighted sum formula of the zeta values at even arguments, and a weighted sum formula of the multiple zeta values with even arguments and its zeta-star analogue. The weight coefficients are given by (symmetric) polynomials of the arguments. These weighted sum formulas for the zeta values and for the multiple zeta values were conjectured by L. Guo, P. Lei and J. Zhao.
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta values, and the Witten zeta function. We discuss a heuristic for finding or dismissing the existence of similar simple sums. We also produce some new sums from recursions involving the Riemann zeta and the Dirichlet beta functions.
A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coeffcients in the rational polynomial ring generated by the two parame…
The sum formula is a well known relation in the field of the multiple zeta values. In this paper, we present its generalization for the Euler-Zagier multiple zeta function.
We prove a sum formula with 4 parameters among finite alternating multiple zeta values which can be regarded as an alternating version of the result of Kamano on finite multiple zeta values.
Hirose, Saito, and the author established the weighted sum formula for finite multiple zeta(-star) values. In this paper, we present its alternative proof. The proof is also valid for symmetric multiple zeta(-star) values.
We give a parameterized generalization of the sum formula for quadruple zeta values. The generalization has four parameters, and is invariant under a cyclic group of order four. By substituting special values for the parameters, we also obtain weighted sum formulas for quadruple zeta values, which contain some known results.
We prove some generalizations of the sum formula for multiple zeta values by using Hiroyuki Ochiai's method of proving the sum formula.
We prove a sum-shuffle formula for multiple zeta values in Tate algebras (in positive characteristic), introduced in \cite{PEL3}.This follows from an analog result for double twisted power sums, implying that an ${\mathbb{F}\_p$-vector space generated by multiple zeta values in Tate algebras is an ${\mathbb{F}\_p$-algebra.
Weighted sum formula for variants of half multiple zeta values
本文利用广义超几何函数${}_{5}F_{4}$的Dougall恒等式，证明了半多重zeta值及其变体的加权和公式，并讨论了插值细化对称多重zeta值的类似结果。
On alternating sum formulas for the Arakawa-Kaneko multiple zeta values
本文构造了Arakawa-Kaneko zeta值交错和的生成函数，证明了其交错和公式，并将该方法推广应用于其他zeta值。
Interpolated multiple $q$-zeta values are deformation of multiple $q$-zeta values with one parameter, $t$, and restore classical multiple zeta values as $t = 0$ and $q \to 1$. In this paper, we discuss generating functions for sum of interpolated multiple $q$-zeta values with fixed weight, depth and $i$-height. The functions are systematically expressed in terms of the basic hypergeometric functions. Compared with the result of Ohno and Zagier, our result includes three generalizations: general height, $q$-deformation and $t$-interpolation. As an application, we prove some expected relations f…
We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.
We introduce an explicit formula for a reciprocal sum related to the Riemann zeta function at s=6, and pose one question related to a computational formula for larger values of s.
The $t$-adic symmetric multiple zeta values were defined Jarossay, which have been studied as a real analogue of $\boldsymbol{p}$-adic finite multiple zeta values. In this paper, we consider the star analogues based on several regularization processes of multiple zeta-star values: harmonic regularization, shuffle regularization, and Kaneko-Yamamoto's type regularization. We also present the cyclic sum formula for $t$-adic symmetric multiple zeta(-star) values, which is the counterpart of that for $\boldsymbol{p}$-adic finite multiple zeta(-star) values obtained by Kawasaki. The proof uses our …
We show that the duality relation for the sum of multiple zeta values with fixed weight, depth and $k_1$ is deduced from the derivation relations, which was first conjectured by N. Kawasaki and T. Tanaka.
$p$-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。
The Ohno relation is a well-known relation among multiple zeta values. Hirose, Onozuka, Sato, and the author investigated the sum related to the Ohno relation and presented two types of new relations and five conjectural formulas. This paper proves one of these formulas.
Lower bounds for high moments of zeta sums
本文研究和式∑_{n≤x}n^{it}的高阶矩，证明了该高阶矩的无条件下界。
Ohno's relation states that a certain sum, which we call an Ohno type sum, of multiple zeta values remains unchanged if we replace the base index by its dual index. In view of Oyama's theorem concerning Ohno type sums of finite and symmetric multiple zeta values, Kaneko looked at Ohno type sums of finite and symmetric multiple zeta-star values and made a conjecture on the generating function for a specific index of depth three. In this paper, we confirm this conjecture and further give a formula for arbitrary indices of depth three.
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.
A recent paper of Furdui and Valean proves some results about sums of products of "tails" of the series for the Riemann zeta function. We show how such results can be proved with weaker hypotheses using multiple zeta values, and also show how they can be generalized to products of three or more such tails.
Large zeta sums
本文研究 t 很大时 zeta 和 $\sum_{n\le x}n^{it}$ 的行为，证明其渐近行为与 Omega 结果，并与大特征和的相关结果进行类比。
A discussion involving the evaluation of the sum $\sum_{0<γ\le T} |ζ(1/2+iγ)|^2$ is presented, where $γ$ denotes imaginary parts of complex zeros of the Riemann zeta-function $ζ(s)$. Three theorems involving certain integrals related to this sum are proved, and the sum is unconditionally shown to be $\ll T\log^2T\log\log T$.
Arithmetic sums and products of infinite multiple zeta-star values
本文研究了受限索引的无穷多重ζ星值的算术和与积，并受连分数理论和康托集启发，提出关于特定索引的ζ星值代数点及算术运算的猜想。
Cyclic sum formula for certain parametrized multiple zeta values
本文针对Igarashi推广Ohno-Wakabayashi多重泽塔星值循环和公式后提出的三参数推广问题之一，给出了可能的解答。
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.
In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By applying these formulas, we can prove that multiple zeta star values whose indices are the sequences $(\bar 1,\{1\}_m,\bar 1)$ and $(2,\{1\}_m,\bar 1)$ can be expressed polynomially in terms of zeta values, polylogarithms and $\ln2$. Finally, we also evaluate several restricted sum formulas involving multiple zeta values.
Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper we prove a family of identities involving Bernoulli numbers and apply them to obtain infinitely many weighted sum formulas for double zeta values and triple zeta values where the weight coefficients are given by symmetric polynomials. We give a general conjecture in arbitrary depth at the end of the paper.
In this paper, we are interested by the cotangent sum c0(q/p) related to the Estermann zeta function for the special case when q = 1 and get explicit formula for its series expansion, which represents an improvement of the identity (2:1) Theorem (2:1) in the recent work [11].
We consider sums of the form $\sum ϕ(γ)$, where $ϕ$ is a given function, and $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such sums can be accelerated by a simple device, and give examples involving both convergent and divergent infinite sums.
Under GRH, we establish a version of Duke's short-sum theorem for entire Artin $L$-functions. This yields corresponding bounds for residues of Dedekind zeta functions. All numerical constants in this work are explicit.
In this paper, we establish some expressions of series involving harmonic numbers and Stirling numbers of the first kind in terms of multiple zeta values, and present some new relationships between multiple zeta values and multiple zeta star values. The relationships obtained allow us to find some nice closed form representations of nonlinear Euler sums through Riemann zeta values and linear sums. Furthermore, we show that the combined sums \[H\left( {a,b;m,p} \right) := \sum\limits_{a + b = m - 1} {ζ\left( {{\left\{ p \right\}_a},p + 1,{\left\{ p \right\}_b}} \right)}\quad (m\in \N,p>1) \] an…
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.
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.
In this paper we use the generating functions and the double shuffle relations satisfied the multiple zeta values to derive some new families of identities.
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.
Refined sum formulas for finite multiple zeta values of level two
本文针对第二层级有限多重zeta值建立了两类精细求和公式，包括一个显式求和公式和一个Ohno--Zagier型关系，并由此推导出固定权重、深度及特殊高度的求和公式。
Bowman and Bradley proved an explicit formula for the sum of multiple zeta values whose indices are the sequence (3,1,3,1,...,3,1) with a number of 2's inserted. Kondo, Saito and Tanaka considered the similar sum of multiple zeta-star values and showed that this value is a rational multiple of a power of π. In this paper, we give an explicit formula for the rational part. In addition, we interpret the result as an identity in the harmonic algebra.
We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cramér, and also new representations for some arithmetical functions in terms of the non-trivial zeros.
An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to prove the identity.
Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the symmetric sum formula, the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations.
Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, …
We prove an asymptotic for the sum of $ζ^{(n)} (ρ)X^ρ$ where $ζ^{(n)} (s)$ denotes the $n$th derivative of the Riemann zeta function, $X$ is a positive real and $ρ$ denotes a non-trivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height $T$, as $T \rightarrow \infty$. We also specify what the asymptotic formula becomes when $X$ is a positive integer, highlighting the differences in the asymptotic expansions as $X$ changes its arithmetic nature.
In this paper, we present some results on the $a$-points of the symmetric sum of the Euler-Zagier multiple zeta function. Our first three results are for the $a$-points free region of the function. The fourth result is the Riemann-von Mangoldt type formula. In the last two results, we study the real parts of $a$-points of the function.
In this paper, we give identities involving cyclic sums of regularized multiple zeta values of depth less than $5$. As a corollary, we present two kinds of extensions of Hoffman's theorem for symmetric sums of multiple zeta values for this case.
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