On a generalization of the cyclic sum formula for finite multiple zeta and zeta-star values

Identities involving cyclic sums of regularized multiple zeta values each of depth less than $5$

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.

Reducibility of Signed Cyclic Sums of Mordell-Tornheim Zeta and L-Values

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.

Note on cyclic sum of certain parametrized multiple series

We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.

The alternating block decomposition of iterated integrals, and cyclic insertion on multiple zeta values

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lisoněk states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ ζ(1,3,\ldots,1,3) $ and summing, one obtains an explicit rational multiple of a power of $ π$. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 ζ(3,3,\{2\}^m) - ζ(3,\{2\}^m,(1,2)) $. In this paper we introduce the 'generalised cyclic insertion conjecture', which we describe using a new combinatorial structure on iterated integrals -- the so-called alternating block decomposition. We see …

Combinatorial remarks on the cyclic sum formula for multiple zeta values

The multiple zeta values are generalizations of the values of the Riemann zeta function at positive integers. They are known to satisfy a number of relations, among which are the cyclic sum formula. The cyclic sum formula can be stratified via linear operators defined by the second and third authors. We give the number of relations belonging to each stratum by combinatorial arguments.

Computing Zeta Functions of Cyclic Covers in Large Characteristic

We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic $p$ that runs in time $p^{1/2 + o(1)}$. We confirm its practicality and effectiveness by reporting on the performance of our SageMath implementation on a range of examples. The algorithm relies on Gonçalves's generalization of Kedlaya's algorithm for cyclic covers, and Harvey's work on Kedlaya's algorithm for large characteristic.

Interpolation of multiple zeta and zeta-star values

We define polynomials of one variable t whose values at t=0 and 1 are the multiple zeta values and the multiple zeta-star values, respectively. We give an application to the two-one conjecture of Ohno-Zudilin, and also prove the cyclic sum formula for these polynomials.

A parameterized generalization of the sum formula for quadruple zeta 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.

Yamamoto's interpolation of finite multiple zeta and zeta-star values

We study a polynomial interpolation of finite multiple zeta and zeta-star values with variable $t$, which is an analogue of interpolated multiple zeta values introduced by Yamamoto. We introduce several relations among them and, in particular, prove the cyclic sum formula, the Bowman-Bradley type formula, and the weighted sum formula. The harmonic relation, the shuffle relation, the duality relation, and the derivation relation are also presented.

A note on evaluations of multiple zeta values

Multiple zeta values (MZVs) with certain repeated arguments or certain sums of cyclically generated MZVs are evaluated as rational multiple of powers of $π^2$. In this paper, we give a short and simple proof of the remarkable evaluations of MZVs established by D. Borman and D. M. Bradley.

On a generalization of restricted sum formula for multiple zeta values and finite multiple zeta values

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.

Generating functions for sums of polynomial multiple zeta values

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.

Weighted sum formulas of multiple zeta values with even arguments

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.

Some Notes on Weighted Sum Formulae for Double Zeta Values

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.

Weighted sums with two parameters of multiple zeta values and their formulas

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…

Sum formula for multiple zeta function

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.

Weighted sum formulas for finite alternating multiple zeta values with some parameters

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.

A restricted sum formula for a q-analogue of multiple zeta values

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.

A combinatorial proof of the weighted sum formula for finite and symmetric multiple zeta(-star) 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.

On generalizations of the sum formula for multiple zeta values

We prove some generalizations of the sum formula for multiple zeta values by using Hiroyuki Ochiai's method of proving the sum formula.

A sum-shuffle formula for zeta values in Tate algebras

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.

半多重zeta值变体的加权和公式

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。

关于Arakawa-Kaneko多重zeta值的交错和公式

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。

Sum of interpolated multiple $q$-zeta values

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…

A reciprocal sum related to the Riemann zeta function at s=6

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.

Derivation relations and duality for the sum of multiple zeta values

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 多重ζ值与二项多重调和和

$p$-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。

Restricted sum formula and derivation relation 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.

A note on Ohno sums for multiple zeta values

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.

Some restricted sum formulas for double zeta values

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.

ζ和的高阶矩的下界

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。

Generating functions for Ohno type sums of finite and symmetric multiple zeta-star values

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.

Finite Multiple zeta Values and Finite Euler Sums

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.

大 zeta 和

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。

On sums of squares of the Riemann zeta-function on the critical line

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 $\ll T\log^2T\log\log T$.

无穷多重ζ星值的算术和与积

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。

Non-vanishing of certain cyclotomic multiple harmonic sums and application to the non-vanishing of certain $p$-adic cyclotomic multiple zeta values

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.

Families of weighted sum formulas for 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.

Series representation of a cotangent sum related to the Estermann zeta function

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].

Accurate estimation of sums over zeros of the Riemann zeta-function

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.

Explicit estimates for Artin $L$-functions: Duke's short-sum theorem and Dedekind Zeta Residues

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.

Multiple zeta values and Euler sums

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…

Shintani's zeta function is not a finite sum of Euler products

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.

Summation rules for the values of the Riemann zeta-function and generalized harmonic numbers obtained using Laurent developments of polygamma functions and their products

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.

New Families of Weighted Sum Formulas for Multiple Zeta Values

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.

On Mordell-Tornheim sums and multiple zeta values

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.

第二层级有限多重zeta值的精细求和公式

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。

Explicit evaluation of certain sums of multiple zeta-star values

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.

Identity involving symmetric sums of regularized multiple zeta-star values

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.

Algebraic relations of interpolated multiple zeta values

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, partial exponential sums, and p-adic estimates

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, …

A further generalisation of sums of higher derivatives of the Riemann Zeta Function

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.

On the $a$-points of symmetric sum of multiple zeta function

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.

Linear relations of Ohno sums of multiple zeta values

Ohno's relation is a well-known family of relations among multiple zeta values, which can naturally be regarded as a type of duality for a certain power series which we call an Ohno sum. In this paper, we investigate $\mathbb{Q}$-linear relations among Ohno sums which are not contained in Ohno's relation. We prove two new families of such relations, and pose several further conjectural families of such relations.