Niranjan Ramachandran
We study the behaviour near s=1/2 of zeta functions of varieties over finite fields F_q with q a square. The main result is an Euler-characteristic formula for the square of the special value at s=1/2. The Euler-characteristic is constructed from the Weil-etale cohomology of a certain supersingular elliptic curve.
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In this paper, we introduce zeta values of rational convex cones, which is a generalization of cyclotomic multiple zeta values. These zeta values have integral expressions. The main theorem asserts that zeta values of cones can be expressed as linear combinations of cyclotomic multiple zeta values over some cyclotomic field.
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 …
In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all the algebraic relations are those algebraic relations within each individual family {zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence resu…
In analogy with values of the classical Euler Gamma-function at rational numbers and the Riemann zeta-function at positive integers, we consider Thakur's geometric Gamma-function evaluated at rational arguments and Carlitz zeta-values at positive integers. We prove that, when considered together, all of the algebraic relations among these special values arise from the standard functional equations of the Gamma-function and from the Euler-Carlitz relations and Frobenius p-th power relations of the zeta-function.
We give a proof of double shuffle relations for $p$-adic multiple zeta values by developing higher dimensional version of tangential base points and discussing a relationship with two (and one) variable $p$-adic multiple polylogarithms.
In this paper we will study the double zeta values $ζ(k,m)$ using Picard-Fuchs equation. We will give a very efficient method to evaluate $ζ(k,1)$ (resp. $ζ(k,2)$) in terms of the products of zeta values $ζ(2),ζ(3),\cdots$ when $k$ is even (resp. odd), which admits immediate generalization to arbitrary double zeta values. Moreover, this method provides new insights into the nature of double zeta values, which further can be generalized to arbitrary multiple zeta values.
An explicit formula for the height-one multiple zeta values was proved by Kaneko and the second author. We give an alternative proof of this result and its generalization. We also prove its counterpart for the 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.
The conical zeta values are a generalization of the multiple zeta values which are defined by certain multiple sums over convex cones. In this paper, we present a relation between the values of the Dedekind zeta functions for totally real fields and the conical zeta values for certain algebraic cones. More precisely, we show that the values of the partial zeta functions for totally real fields can be expressed as a rational linear combination of the conical zeta values associated with certain algebraic cones up to the square root of the discriminant.
We define the interpolated polynomial multiple zeta values as a generalization of all of multiple zeta values, multiple zeta-star values, interpolated multiple zeta values, symmetric multiple zeta values, and polynomial multiple zeta values. We then compute the generating function of the sum of interpolated polynomial multiple zeta values of fixed weight, depth, and height.
Cyclotomic symmetric multiple zeta values span the space of cyclotomic multiple zeta values
该研究证明了不同学者独立提出的分圆对称多重zeta值可张成模$\pi i$的分圆多重zeta值空间，为分圆多重zeta值的结构研究提供了关键结论。
In this paper we consider iterated integrals of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst \cite{BBBL1997}, and give some general results. Furthermore, we discuss Kaneko-Yamamoto multiple zeta values, and establish some relations between it and multiple zeta values. Finally, we establish a linear re…
The derivation relation is a well known relation among multiple zeta values, which was first obtained by Ihara, Kaneko and Zagier. The analogous formula for finite multiple zeta values, which we call the derivation relation for finite multiple zeta values, was conjectured by the third author and proved by the second author. In this paper, we show these two kinds of derivation relations are respectively equivalent to the Ohno type relations for multiple zeta values and finite multiple zeta values. We also reprove these derivation relations in several different ways.
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.
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.
In this paper we define a continuous version of multiple zeta functions. They can be analytically continued to meromorphic functions on $\mathbb{C}^r$ with only simple poles at some special hyperplanes. The evaluations of these functions at positive integers (continuous multiple zeta values) satisfy the shuffle product. We give a detailed analysis about the depth structure of continuous multiple zeta values. There are also sum formulas for continuous multiple zeta values. Lastly we calculate some special continuous multiple zeta values in terms of special values of multiple polylogarithms.
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.
Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing the integral kernel associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.
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 first review our previous works of Arakawa and the authors on two, closely related single-variable zeta functions. Their special values at positive and negative integer arguments are respectively multiple zeta values and poly-Bernoulli numbers. We then introduce, as a generalization of Sasaki's work, level 2 analogue of one of the two zeta functions and prove results analogous to those by Arakawa and the first named author.
The derivation relations for multiple zeta values is proved by Ihara, Kaneko and Zagier. We prove its counterpart for finite multiple zeta values.
There has been an avalanche of recent research on multiple zeta values. We propose dividing identities for multiple zeta values into structural and specific types. Structural identities are valid for any generalized multiple zeta function, and we systematically investigate them through symmetric functions. Specific identities are only valid for a specific zeta function, and we show how these can be used in conjunction with structural identities to find closed form multiple zeta values. This allows us to interpret generalized multiple zeta values as the moments of a random variable, which we ch…
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.
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 prove the Ohno-type relation for the interpolated multiple zeta values, which was introduced first by Yamamoto. Same type results for finite multiple zeta values are also given. Moreover, these relations give the sum formula for interpolated multiple zeta values and interpolated $\mathcal{F}$-multiple zeta values, which were proved by Yamamoto and Seki, respectively.
We prove a kind of integral expressions for finite multiple harmonic sums and multiple zeta-star values. Moreover, we introduce a class of multiple integrals, associated with some combinatorial data (called 2-labeled posets). This class includes both multiple zeta and zeta-star values of Euler-Zagier type, and also several other types of multiple zeta values. We show that these integrals can be used to obtain some relations among such zeta values quite transparently.
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.
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。
Nakasuji, Phuksuwan, and Yamasaki defined the Schur multiple zeta values and gave iterated integral expressions of the Schur multiple zeta values of the ribbon type. This paper generalizes their integral expressions to the ones of more general Schur multiple zeta values having constant entries on the diagonals. Furthermore, we also discuss the duality relations for Schur multiple zeta values obtained from the integral expressions.
We study some classical identities for multiple zeta values and show that they still hold for zeta functions built on the zeros of an arbitrary function. We introduce the complementary zeta function of a system, which naturally occurs when lifting identities for multiple zeta values to identities for quasisymmetric functions.
We study a variant of multiple zeta values of level 2, which forms a subspace of the space of alternating multiple zeta values. This variant, which is regarded as the `shuffle counterpart' of Hoffman's `odd variant', exhibits nice properties such as duality, shuffle product, parity results, etc., like ordinary multiple zeta values. We also give some conjectures on relations between our values, Hoffman's values, and multiple zeta values.
Higher-genus multiple zeta values
该研究将多重zeta值的形式体系从经典及椭圆情形规范扩展到高亏格黎曼曲面，提出基于肖特基单值化的正则化方案，探究几何退化和多重对数关系衍生的高亏格多重zeta值关系，并通过数值验证发现了新的关系类别。
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.
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.
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 give three identities involving multiple zeta values of height one and of maximal height; an explicit formula for the height-one multiple zeta values, a regularized sum formula, and a sum formula for the multiple zeta values of maximal height.
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.
We prove that the $\mathbb{Q}$-vector space generated by the multiple zeta values is generated by the finite real multiple zeta values introduced by Kaneko and Zagier.
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…
We prove three theorems on finite real multiple zeta values: the symmetric formula, the sum formula and the height-one duality theorem. These are analogues of their counterparts on finite multiple zeta values.
This work is an example driven overview article of recent works on the connection of multiple zeta values, modular forms and q-analogues of multiple zeta values given by multiple Eisenstein series.
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.
Ohno's relation is a well-known relation among multiple zeta values.In this paper, we prove Ohno-type relation for finite multiple zeta values, which is conjectured by Kaneko.As a corollary, we give an alternative proof of the sum formula for finite multiple zeta values, which was first proved by Saito and Wakabayashi.
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.
In this paper, we investigate three general forms of multiple zeta(-star) values. We use these values to give three new sum formulas for multiple zeta(-star) values with height $\leq 2$ and the evaluation of $ζ^\star(\{1\}^m,\{2\}^{n+1})$. We also give a new proof of sum formula of multiple zeta values.
We prove an easy but interesting result about the linear independence of multiple zeta values of different weights.
In this paper, we introduce the method of adding additional factors and a parameter to multiple zeta values and prove some generalizations of the duality theorem and several relations among multiple zeta values. In particular, we are able to evaluate some special (truncated) sums in terms of Riemann zeta values of different weights.
Regularization of elliptic multiple zeta values
本文证明正则化椭圆多重zeta值可表示为可容许指标与特殊指标（仅含0和1）的椭圆多重zeta值的多项式。
In this paper, the extended double shuffle relations for interpolated multiple zeta values are established. As an application, Hoffman's relations for interpolated multiple zeta values are proved. Furthermore, a generating function for sums of interpolated multiple zeta values of fixed weight, depth and height is represented by hypergeometric functions, and we discuss some special cases.
We present several conjectures on multiple q-zeta values and on the role they play in certain problems of enumerative geometry.
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。
We establish a short exact sequence about depth-graded motivic double zeta values of even weight relative to $μ_2$. We find a basis for the depth-graded motivic double zeta values relative to $μ_2$ of even weight and a basis for the depth-graded motivic triple zeta values relative to $μ_2$ of odd weight. As an application of our main results, we prove Kaneko and Tasaka's conjectures about the sum odd double zeta values and the classical double zeta values. We also prove an analogue of Kaneko and Tasaka's conjecture in depth three. At last we formulate a conjecture which is related to sum odd m…
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.
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 give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.
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.
In this article, we introduce an algebraic setup of non-strict multiple zeta values (NMZVs, for short) and prove some relations of NMZVs, which are analogous to Hoffman's relations of multiple zeta values, by using this algebraic setup of NMZVs.
For several evaluations of special values and several relations known only in $\mathcal{A}_n$-multiple zeta values or $\mathcal{S}_n$-multiple zeta values, we prove that they are uniformly valid in $\mathcal{F}_n$-multiple zeta values for both the case where $\mathcal{F}=\mathcal{A}$ and $\mathcal{F}=\mathcal{S}$. In particular, the Bowman-Bradley type theorem and sum formulas for $\mathcal{S}_2$-multiple zeta values are proved.
Ihara, Kaneko, and Zagier proved the derivation relation for multiple zeta values. The first named author obtained its counterpart for finite multiple zeta values in $\mathcal{A}$. In this paper, we present its generalization in $\mathcal{A}_{n}$.
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.
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