Diophantine problems for q-zeta values
Wadim Zudilin
给出了黎曼ζ函数值在正整数上的一个有动机的q-扩张.几个非理性和超越性的结果，以及新的一般问题，这些q-zeta值的陈述。
A motivated q-extension of the values of the Riemann zeta function at positive integers is presented. Several irrationality and transcendence results as well as new general problems for these q-zeta values are stated.
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Diophantine approximation of multiple zeta-star values
本文针对多重zeta星值集合建立了丢番图逼近的经典结果，包括判定准则、零一律及一个加强的猜想。
Hankel Determinants of Zeta Values
本文研究了在等差数列中由黎曼ζ函数在正整数处的值$ζ(an+b)$构造的Hankel行列式的渐近性。我们的主要结果是一个丢番图应用的渐近性。
On Diophantine properties for values of Dedekind zeta functions
本文研究Dedekind zeta函数特殊值的Northcott与Bogomolov性质，证明当σ≥1/2时Bogomolov性质不成立，并构造数域族使zeta函数值任意小，同时推广了Northcott性质至Re(s)<0的情形。
Aspects of Zeta-Function Theory in the Mathematical Works of Adolf Hurwitz
阿道夫赫尔维茨是相当著名的贡献，黎曼曲面，模形式，丢番图方程和近似以及某些方面的代数。他的早期工作的一个重要推广狄利克雷的$L$系列，现在称为赫维茨ζ函数，是唯一出版的工作解决了非常活跃的研究领域周围的黎曼ζ函数及其亲属。他的数学日记，但是，提供了另一种情况，即终身兴趣的发展zeta函数理论。在本说明中，我们将调查他的早期工作，它的起源和它的接待，以及赫尔维茨的进一步研究的黎曼ζ函数和盟军狄利克雷级数从他的日记。事实证明，赫尔维茨已经在1889年知道的基本分析性质的爱泼斯坦zeta函数（包括其函数方程）13年前保罗爱泼斯坦。
In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In fact, in one of the famous article, Zagier obtained an explicit formula for Dedekind zeta function at point 2 and conjectured an identity at any even values over any number field. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even a…
Sum relations for multiple zeta values and connection formulas for the Gauss hypergeometric functions
我们给出了一个明确的表示为多个zeta-star值的固定重量和高度的Riemann zeta值的总和。
A Mean Value Theorem for the Diophantine Equation $axy-x-y=n$
本文证明了丢番图方程$axy-x-y=n$的平均解数的一个渐近公式，其中$a$是固定的，而$n$是变化的.
Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form $P_{A}(x)=P_{B}(y)$, where $A, B$ are certain finite sets.
Finiteness results for Diophantine triples with repdigit values
令$g\ge 2$为整数，并且$\mathcal R_g\subset \mathbb N$为基$g$中的repdigits的集合。令$\mathcal D_g$是具有$\mathcal R_g$中的值的丢番图三元组的集合;即，$\mathcal D_g$是具有$c<b<a$的所有三元组$(a,b,c)\in \mathbb N^3$的集合，使得$ab+1,ac+1$和$ab+1$位于集合$\mathcal R_g$中。本文证明了集合$\mathcal D_g$的有效有限性结果。
On the spectrum of Diophantine exponents of lattices
本文研究了任意维数格点的弱均匀丢番图指数的谱特性。
On multiple zeta values and finite multiple zeta values of maximal height
Kaneko和第二作者证明了高度1倍zeta值的显式公式。我们给出了这个结果的另一种证明及其推广。我们也证明了它的对应有限多个zeta值。
We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value in terms of multiple alternating Euler sums, and specifically, reduce the length of blocks of twos in the resulting sums.
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。
We prove integral representations of the approximation forms in zeta values constructed in arXiv:1801.09895 and arXiv:1803.08905.
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.
Interpolation of multiple zeta and zeta-star values
我们定义一个变量t的多项式，其在t=0和1的值分别是多个zeta值和多个zeta-star值。我们给出了Ohno-Zudilin的二-一猜想的一个应用，并证明了这些多项式的循环和公式。
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.
Large values of L-functions from the Selberg class
本文证明了Selberg类中L函数的下界，从而改进了第二作者与Jörn Steuding的早期结果.我们制定了两个定理，使用略有不同的技术假设，并给出了两个完全不同的证明。第一个证明使用的“共振法”，这是介绍了Soundararajan，而第二个证明使用的方法从丢番图逼近这类似于那些使用蒙哥马利。有趣的是，这两种方法导致大致相同的下界，这是不符合那些已知的黎曼ζ函数，似乎很难被改善。除了这些结果之外，我们还证明了Selberg类中L函数的上界，并给出了Chen的一个定理的进一步应用，该定理用于上述的丢番图逼近方法.
Mean Values over Lattices in Number Fields and Effective Diophantine Approximation
本文利用 Rogers 平均值定理在代数数域上的推广及 Gaposhkin 有效遍历定理，解决了 Alam 和 Ghosh 关于数域有理逼近螺旋结果误差项的问题。
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.
Recently, several new results related to the evaluation of the series sum (-1)^n zeta(n)/(n+k) were published. In this short note we show that this series also possesses an interesting connection to the values of the zeta-function on the critical line and to the Euler constant.
A Note on the Quartic Diophantine Equation $A^4+hB^4=C^4+hD^4$
丢番图方程$A^4+hB^4=C^4+hD^4$的整数解对于$h < 1000$的所有正整数值都是已知的。虽然对于$h$的任意正整数值的上述丢番图方程的解是未知的，但Gerardin和Piezas发现了当$h$分别由5次和2次多项式给出时该方程的解。本文给出了当$h$由$2,\;3$和4次多项式给出时该方程的几种新解。
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.
In this note we investigate the set $S(n)$ of positive integer solutions of the title Diophantine equation. In particular, for a given $n$ we prove boundedness of the number of solutions, give precise upper bound on the common value of $σ_{2}(\overline{X}_{n})$ and $σ_{n}(\overline{X}_{n})$ together with the biggest value of the variable $x_{n}$ appearing in the solution. Moreover, we enumerate all solutions for $n\leq 16$ and discuss the set of values of $x_{n}/x_{n-1}$ over elements of $S(n)$.
Is the quartic Diophantine equation $A^4+hB^4=C^4+hD^4$ solvable for any integer $h$?
丢番图方程$A^4+hB^4=C^4+hD^4$，其中$h$是一个固定的任意正整数，已被一些作者研究过.目前，通过计算机搜索，对于$h \le 5000$和$A, B, C, D \le 100000$的所有正整数值，除了一些数字之外，该方程的整数解是已知的，而对于$h$的任意正整数值，该丢番图方程的解是未知的。Gerardin和Piezas分别用$5$和$2$的次数多项式给出了$h$时，找到了该方程的解。当$h$由$2$、$3$和$4$的次数多项式给出时，Choudhry也给出了该方程的一些新解.本文利用椭圆曲线理论研究了这类丢番图方程，其中$h$是一个固定的任意有理数.本文给出了某些$h$值的丢番图方程的一些解，特别是对于那些在$A, B, C, D \le 100000$的范围内还没有找到解的值.当$h$由$3$，$4$的次数多项式给出时，给出了丢番图方程的一些新的参数解.最后给出了两种解法，只要其中一种正确，就可以求解任意有理数$h$的上述丢番图方程.
Diophantine methods for exponential sums, and exponential sums for Diophantine problems
讨论了Hardy-Littlewood方法在理论和应用方面的最新进展，重点讨论了对角丢番图问题。最近有效的差分方法估计平均值的指数和第一，专注于发展涉及光滑外尔和。其次，讨论了经典的贝塞尔不等式和柯西-施瓦茨不等式的算术变体。最后，一些新兴的圆的方法和算术几何之间的联系被提及。
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.
An algebraic proof of cyclic sum formula for multiple zeta values
我们介绍了一个代数制定的循环和公式的多个zeta值和多个zeta星值。我们还提出了一个代数证明的循环和公式的多个zeta值和多zeta星值，通过减少他们的川岛关系。
Diophantine approximation with arithmetic functions, I
本文证明了一个强的同时丢番图逼近定理，它适用于加法函数和乘性函数的值，只要这两类函数在素数上具有一定的正则性。
Extreme value laws for intrinsic Diophantine approximation on spheres
本文研究球面上内在丢番图逼近的极值行为，证明在稀疏高度序列下逼近质量服从Weibull极限分布，计数函数渐近泊松。
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.
Derivation relations for finite multiple zeta values
Ihara，Kaneko和Zagier证明了多重zeta值的导子关系。我们证明其对应的有限多个zeta值。
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.
Some new results in quantitative Diophantine approximation
本文针对对角三元不定形式的一参数族，建立了两个定量丢番图逼近的新结果，分别涉及素数点上的二次形式与整数点上的任意次数非齐次形式。
Multiple zeta-star values and multiple integrals
证明了有限重调和和与重zeta-星值的一类整式。此外，我们还引入了一类与组合数据相关的多重积分（称为2-标号偏序集）。该类别包括Euler-Zagier类型的多重zeta和zeta星值，以及其他几种类型的多重zeta值。我们表明，这些积分可以用来获得这样的zeta值相当透明的一些关系。
Shuffle product formulas of multiple zeta values
利用混洗积的组合描述，证明或重新表述了几个多重zeta值的混洗积公式，包括两个多重zeta值的混洗积的一般公式，两个多重zeta值的乘积的限制性混洗积公式，以及$n$多重zeta值的乘积的限制性混洗积公式.
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。
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.
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.
Identities for the multiple zeta (star) values
本文利用对数函数的积分计算和级数的迭代整表示的方法，证明了任意深度的多重zeta值和多重zeta星值的一些新的恒等式。应用这些公式，我们可以证明，其指数为序列$(\bar 1,\{1\}_m,\bar 1)$和$(2,\{1\}_m,\bar 1)$的多重zeta星值可以用zeta值、多项式和$\ln2$多项式表示。最后，我们还评估了几个限制和公式涉及多个zeta值。
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 …
On multiple zeta values of extremal height
我们给三个身份涉及多个zeta值的高度1和最大高度;一个明确的公式高度1多个zeta值，正则化的总和公式，和总和公式的多个zeta值的最大高度。
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.
Finite real multiple zeta values generate the whole space $Z$
证明了由多重zeta值生成的$\mathbb{Q}$ -向量空间是由Kaneko和Zagier引入的有限实多重zeta值生成的。
Weighted sums with two parameters of multiple zeta values and their formulas
多个zeta值的典型公式是求和公式，其将黎曼zeta值表示为固定重量和深度的所有多个zeta值的和。加权和公式是和公式的加权类似，近年来得到了广泛的研究。本文给出了多个zeta值的两个含两个参数的加权和公式。作为公式的应用，我们找到了一些多个zeta值的线性组合，它们可以表示为系数在由两个参数生成的有理多项式环中的通常zeta值的多项式，并得到了小深度的多个zeta值的加权和的一些恒等式。
A Note on Finite Real Multiple Zeta Values
证明了有限实重zeta值的三个定理：对称公式、求和公式和高一对偶定理。这些是它们在有限多个zeta值上的对应物的类似物。
Multiple Eisenstein series and q-analogues of multiple zeta values
这项工作是一个例子驱动的概述文章最近的作品连接的多个zeta值，模形式和q-类似物的多个zeta值给出多个爱森斯坦系列。
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-type relation for finite multiple zeta values
Ohno关系是多重zeta值之间的一个著名关系，本文证明了Kaneko证明的有限多重zeta值的Ohno型关系，作为推论，给出了Saito和Wakabayashi首先证明的有限多重zeta值求和公式的另一种证明.
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.
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