Note on relations among multiple zeta(-star) values II

关于黎曼ζ函数四次幂矩的一个注记

A note on the fourth power moment of the Riemann zeta-function

给出了黎曼ζ函数在临界线上的平均四次幂的渐近展开式中各项的显式表达式。

Euler-Barnes多重Zeta函数的q-表示（=关于q-zeta函数的一个注记）

q-Analogue of Euler-Barnes Multiple Zeta Functions (= a note on q-zeta functions)

最近（见[1]），我介绍了一个有趣的Euler-Barnes多重zeta函数。本文构造了Euler-Barnes多重zeta函数的q-模拟，它是高阶Frobenius-欧拉数的q-模拟在负整数处的插值.

A note on derivation relations for multiple zeta values and finite multiple zeta values

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.

A note on odd zeta values

Using a new construction of rational linear forms in odd zeta values and the saddle point method, we prove the existence of at least two irrational numbers amongst the 33 odd zeta values $ζ$(5), $ζ$(7),. .. , $ζ$(69).

关于双Zeta值加权和公式的几点注记

Some Notes on Weighted Sum Formulae for Double Zeta Values

我们提出了一个统一的方法，给出了完全初等证明的三个加权和公式的双重zeta值。这种方法也导致新的评价有关的谐波数，交替双zeta值，和威滕zeta函数的总和。我们讨论了一个启发式的发现或驳回类似的简单和的存在。我们还从涉及Riemann zeta和Dirichlet β函数的递归中得到了一些新的和。

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.

Notes on congruence zeta functions via a Berkovich approach

Notes on congruence zeta functions via a Berkovich approach

本文在Scholze的Berkovich动机框架下，利用范畴迹构造了一种新的zeta函数，并证明其与经典同余zeta函数一致。

多重q-欧拉zeta函数与高阶q-欧拉多项式对称性的注记

A note on symmetric properties of the multiple q-Euler zeta functions and higher-order q-Euler polynomials

最近，T。Kim（[8，9]）.本文研究了多重q-欧拉zeta函数的一些对称性质，并由多重q-欧拉zeta函数的对称性质推导出有关高阶q-欧拉多项式的各种等式。

关于Hurwitz-Lerch zeta函数普适性的注记

A note on the universality of Hurwitz-Lerch zeta functions

证明Lerch zeta函数$L(λ,α,s)$在$λ$是无理数且$α$是有理数时以及在$α$是无理代数数时对任何$λ$的普适性的尝试失败。

关于多重zeta值计算的注记

A note on evaluations of multiple zeta values

具有某些重复变元的多个zeta值（MZV）或循环生成的MZV的某些总和被评估为$π^2$的幂的有理数倍。本文对D.博尔曼和D. M.布拉德利。

A Note on Harmonic number identities, Stirling series and multiple zeta values

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.

A note on $\mathcal{F}_n$-multiple zeta values

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.

关于zeta函数和L函数正性条件的注记

A note on some positivity conditions related to zeta- and L-functions

L.德·布兰热提出了一种使用某些正性条件来研究黎曼假设的方法。在本文中，作者研究这种方法，并指出其困难。

关于Okounkov关于多重zeta值的q-模拟的一个猜想的注记

A short note on a conjecture of Okounkov about a q-analogue of multiple zeta values

在[OK] Okounkov研究了一个特定的$q$ -模拟多个zeta值，并作出了一些说明其代数结构。本文将Okounkovs $q$ -类似物与文献[BK 1]中定义的多因子和的生成函数进行比较。我们还提出了一个关于它们的维数的猜想，补充了Okounkov的维数公式，并给出了一些数值证据。

关于黎曼 zeta 函数 q-模拟的阶估计的一个注记

A note on order estimates of the $q$-analogue of the Riemann zeta function

本文针对固定参数 q，首次给出黎曼 zeta 函数 q-模拟在竖直线上的阶估计界，为后续研究奠定基础。

A note on the zeros of the zeta-function of Riemann

It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.

关于S（T）与黎曼ζ函数零点的注记

A note on S(T) and the zeros of the Riemann zeta-function

令$πS(t)$表示黎曼ζ函数在点$\frac12+it$的辐角。假设黎曼假设，我们锐化常数在目前已知的最好的界限为$S(t)$和$S(t)$的变化的时间间隔。然后，我们推导出的zeta函数的零的最大多重性和零之间的最大差距的估计。

关于黎曼ζ函数的零点间隔的一个注记

A note on the gaps between consecutive zeros of the Riemann zeta-function

假设的黎曼假设，我们表明，无限经常连续的非平凡零的黎曼ζ函数相差最多0.5155倍的平均间距和无限经常他们相差至少2.69倍的平均间距。

关于（h，q）-Zeta型函数的一个注记

A note on the (h,q)-Zeta type function with weight alpha

本文的目的是导出带权α的（h，q）-Zeta函数的对称性。利用这一性质，我们给出了带权α的（h，q）-Genocchi多项式的一些有趣的恒等式.因此，我们的应用程序具有一些有趣的属性，我们在本文中陈述。

A note on $r$-gaps between zeros of the Riemann zeta-function

In this note, we prove Selberg's announced result on $r$-gaps between zeros of the Riemann zeta-function $ζ$. Our proof uses a result on variations of $\argζ$ by Tsang based on Selberg's method. The same result with explicit constants under the Riemann Hypothesis has been obtained by Conrey and Turnage-Butterbaugh using a different method. We explain how to obtain explicit constants under the Riemann Hypothesis using our approach which is based on Selberg's and Tsang's arguments.

A note on Dedekind zeta values at 1/2

For a number field $K$, let $ζ_{K}(s)$ be the Dedekind zeta function associated to $K$. In this note, we study non-vanishing and transcendence of $ζ_{K}$ as well as its derivative $ζ_{K}'$ at $s= 1/2$. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of $e^γ$ and non-vanishing of $L$-series at $s= 1/2$, J. Number Theory 161 (2016) 444-456].

A note on the maximum of the Riemann zeta function on the 1-line

We investigate the relationship between the maximum of the zeta function on the 1-line and the maximal order of $S(t)$, the error term in the number of zeros up to height $t$. We show that the conjectured upper bounds on $S(t)$ along with the Riemann hypothesis imply a conjecture of Littlewood that $\max_{t\in [1,T]}|ζ(1+it)|\sim e^γ\log\log T$. The relationship in the region $1/2<σ<1$ is also investigated.

关于多重ζ函数及其变体在相同参数下的注记

A note on the multiple zeta functions and their variants at identical arguments

本文研究多重ζ函数及其变体在相同参数下的表达式，利用调和积将其表示为黎曼ζ函数的多项式，通过贝尔多项式描述系数，并建立多重t函数及其星变体之间的函数关系。

关于临界线上Epstein zeta函数零点间隙的注记

A note on the gaps between zeros of Epstein's zeta-functions on the critical line

证明了与定积分二元二次型有关的Epstein zeta函数$ζ_{Q}(s)$，对充分大的正数$T$，有一个零$1/2 + iγ$且有$ T \leq γ\leq T + T^{{3/7} +\varepsilon} $.这是M. Jutila和K. Srinivas（Bull. London Math.Soc.37（2005）45- 53）。

A Note on Number Fields Sharing the List of Dedekind Zeta-Functions of Abelian Extensions with some Applications towards the Neukirch-Uchida Theorem

Given a number field $K$ one associates to it the set $Λ_K$ of Dedekind zeta-functions of finite abelian extensions of $K$. In this short note we present a proof of the following Theorem: for any number field $K$ the set $Λ_K$ determines the isomorphism class of $K$. This means that if for any number field $K'$ the two sets $Λ_K$ and $Λ_{K'}$ coincide, then $K \simeq K'$. As a consequence of this fact we deduce an alternative approach towards the proof of Neukirch-Uchida Theorem for the case of non-normal extensions of number fields.

A note on the zeros of generalized Hurwitz zeta functions

Given a function $f(n)$ periodic of period $q\geq 1$ and an irrational number $0<α\leq 1$, Chatterjee and Gun proved that the series $F(s,f,α)=\sum_{n=0}^{\infty}\frac{f(n)}{(n+α)^s}$ has infinitely many zeros for $σ>1$ when $α$ is transcendental and $F(s,f,α)$ has a pole at $s=1$, or when $α$ is algebraic irrational and $c=\frac{\max{f(n)}}{\min{f(n)}}<1.15$. In this note, we prove that the result holds in full generality.

关于zeta函数和$L$ -函数零点的一个注记

A note on the zeros of zeta and $L$-functions

令$πS(t)$表示黎曼ζ函数在点$s=\tfrac12+it$处的辐角。在黎曼假设下，给出了$S(t)$已知最尖锐界的一个新的简单证明.我们讨论了这一界的一个大类的$L$ -功能，包括那些所产生的尖形自守表示GL（$m$）在数域上的推广。我们还证明了一些相关的结果，包括边界的消失顺序的$L$ -功能的中心点和边界的高度最低零点的$L$ -功能。

关于黎曼ζ函数均方值的一个注记

A note on the mean square of the Riemann zeta-function

本文针对黎曼ζ函数均方值的已知结果，提出了一种新的证明方法。

A note on small gaps between zeros of the Riemann zeta-function

Assuming the Riemann Hypothesis, we improve on previous results by proving there are infinitely many zeros of the Riemann zeta-function whose differences are smaller than 0.50412 times the average spacing. To obtain this result, we generalize a set of weights that were developed by Xiaosheng Wu, who used them to find a positive proportion of large and small gaps between zeros of the Riemann zeta-function.

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.

关于zeta（3）与Catalan常数乘积的注记

A note on products involving zeta(3) and Catalan's constant

在最近的一份文件卡奇和Tzerkirt给初等证明四个产品公式涉及zeta（3），π，和加泰罗尼亚的常数。他们表示，他们无法直接从Borwein和Dykshoorn于1993年引入的函数值推导出这些产品。我们在这里提供了这样的两个公式的证明。另外两个公式的直接证明是利用Adamchik的Borwein-Dykshoorn函数的推广。最后给出了Borwein-Dykshoorn函数在Xia于2013年提出的“参数化欧拉常数函数”下的表达式，它恰好是K.和T. 2010年的Hessami Pilehrood。

A note on the number of irrational odd zeta values

It is proved that, for all odd integer $s \geqslant s_0(\varepsilon)$, there are at least $\big( c_0 - \varepsilon \big) \frac{s^{1/2}}{(\log s)^{1/2}} $ many irrational numbers among the following odd zeta values: $ζ(3),ζ(5),ζ(7),\cdots,ζ(s)$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. The work is based on the previous work of Fischler, Sprang and Zudilin [FSZ19], improves the lower bound $2^{(1-\varepsilon)\frac{\log s}{\log\log s}}$ therein. The main new ingredient is an optimal design for the zeros of the auxiliary rational functions, which relates to the inverse …

关于Gonek猜想的一个注记

A note on a conjecture of Gonek

我们推导出一个二阶矩的倒数的导数的黎曼ζ函数平均超过零点的ζ函数，这是一半的大小的限制值。我们的结果是以黎曼假设和zeta函数的零点是单的猜想为条件的。

关于黎曼ζ函数实部的一个注记

A note on the real part of the Riemann zeta-function

我们考虑半平面$\Re(s) \ge 1$中黎曼ζ函数$ζ(s)$的实部$\Re(ζ(s))$。我们展示了如何精确地计算常数$σ_0 = 1.19\ldots$，它被定义为$σ$的上确界，使得对于某些实$t$，$\Re(ζ(σ+it))$可以为负（或零）。我们还考虑了$\Re(ζ(1+it)) \le 0$的间隔，并表明它们是罕见的。第一个发生在$t$约682112.9，长度约为0.05。我们列出了前50个这样的间隔。

关于与 Dedekind zeta 函数相关的单零点的一个注记

A note on simple zeros related to Dedekind zeta functions

本文在 $ζ(s)$ 与 $L(s,χ)$ 的 $L^{6}$ 平均 Lindelöf 假设下，证明二次数域 Dedekind zeta 函数非平凡单零点数量的新下界，并加强和改进已有条件性结果。

Algebraic Relations among Goss's Zeta Values on Elliptic Curves

In 2007 Chang and Yu determined all the algebraic relations among Goss's zeta values for the rational function field - these are also known as the Carlitz zeta values. Goss raised the problem about algebraic relations among Goss's zeta values for a general base ring A, but very little is known. In this paper we develop a general method and determine all algebraic relations among Goss's zeta values for the base ring A attached to an elliptic curve over a finite field. To our knowledge, these are the first non-trivial solutions of Goss's problem for a base ring whose class number is strictly gre…

关于无理奇zeta值数量的注记，II

A note on the number of irrational odd zeta values, II

本研究通过结合Fischler-Sprang-Zudilin消元技术与Zudilin的Φₙ因子方法，改进了奇zeta值中无理数数量的下界，将此前结果以常数因子提升至至少1.284·√(s/log s)个。

关于zeta函数和L函数的平均值的注记。XV

A note on the mean value of the zeta and L-functions. XV

本文的目的是使自守$L$ -函数的平均值的谱理论-在一个统一的方式。这是从第十二部分和第十四部分开始的调查的结果，在这两部分中，以自守表示理论为基础建立了一个框架，并设想了一种平均值的一般方法。仅仅为了简单起见，我们将自己限制在由全模群提供的情形。细节和扩展将在其他地方发布。在这个新的版本中，我们增加了我们的论点的各个方面的细节，特别是对离散序列和连续谱的处理。

On primeness of the Selberg zeta-function

In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition.

寻找交替ζ函数导数之间的近似多项式依赖

In Search of Approximate Polynomial Dependencies Among the Derivatives of the Alternating Zeta Function

本文通过数值证据探讨交替ζ函数及其导数之间的近似多项式依赖关系，并提出相关猜想。

Cotype zeta function for subrings of ${\mathbb Z}[t]/(t^3)$

In this note we study the cotypes of subrings of ${\mathbb Z}[t]/(t^3)$ using $p$-adic integration.

关于zeta函数和L函数的平均值的注记。XIV

A note on the mean value of the zeta and L-functions. XIV

本说明的目的是研究自守L函数平均值统一理论的可行性，这是该领域的一个迫切需要。这是从本系列第十二部分开始的研究的结果，在第十二部分中，以自守表示理论为基础建立了一个框架，并设想了一种平均值的一般方法。具体地说，它表明，在这里，内积方法，这是由A。很好，大大增强了M。Jutila，应该被带到完善，使平均平方的L-函数附加到任何尖点形式上半平面可以达到的概念内的自同构。基里洛夫地图是我们的关键工具。由于它的几何性质，我们的论证似乎可以扩展到更大的线性李群。本文件基本上是自成一体的。

关于Zeta函数的一些常数及其与Gregory系数的关系的注记

A note on some constants related to the zeta-function and their relationship with the Gregory coefficients

本文得到了第一、第二Stieltjes常数（也称为广义欧拉常数）以及一些密切相关的常数的新级数。这些级数只包含有理数项，并涉及所谓的格雷戈里系数，也被称为（倒数）对数，第一类柯西数和第二类伯努利数。此外，还给出了Euler常数和常数ln（2*pi）的两个有理项的有趣级数，并提出了Euler常数的另一种推广，得到了计算这些常数的各种公式。最后，在本文中，我们提到，几乎所有的常数在这项工作中考虑承认通过拉马努金求和简单的表示。

多重zeta值之间的一类关系

A class of relations among multiple zeta values

证明了一类新的多重zeta值之间的关系，其中包含Ohno关系。在新的关系式下，我们还给出了固定权的独立MZV的最大数目的公式。为了推导MZV的公式，我们考虑牛顿级数，其值在非负整数是有限的多重调和和。

关于Zeta函数和黎曼假设的几点注记

Some notes about the Zeta function and the Riemann hypothesis

本文的目的是调查是否和如何进行代数分析的Zeta函数和黎曼假设可以进行。当然，在150多年的世界范围内的研究中深入探索的Zeta函数的良好特性是理所当然的。所选择的方法开始于对定义为Re（s）= X < 1的Zeta函数的扩展进行公式化的公认必要性。一个特定形式的扩展，基于欧拉-麦克劳林积分程序，选择和代数评价是由所需的条件，使表达这种扩展等于零。对Zeta函数的这种扩展施加零条件意味着具有不同性质的三项之和为零。只有当满足三项之间的某些相容性条件时，才能得到这一点。其中一项被赋予关于临界线（C.L.）的高度对称性。X = 1/2，从相容性条件可以得出，要么其他两项具有相同类型的对称性，要么任何复共轭（c.c.）一对零点应位于C. L上。由于首项的对称性不为其他两项所共有，因此似乎唯一允许的c. c.零点对必须属于C.L.

关于广义Hurwitz zeta函数的零点

On the zeros of generalized Hurwitz zeta functions

本文证明了一类广义Hurwitzzeta函数在绝对收敛域中存在无穷多个零点。这是Davenport，Heilbronn和Cassels关于Hurwitz zeta函数零点的一个经典问题的推广。

关于黎曼ζ函数共振法的注记

Note on the resonance method for the Riemann zeta function

我们改进Montgomery的$Ω$ -结果$|ζ(σ+it)|$在带$1/2<σ<1$，并特别给出了最大值的$|ζ(σ+it)|$在$\sqrt{T}\le t \le T$是均匀的$σ$的下界。我们给出了类似的最大值的$|\sum_{n\le x} n^{-1/2-it}|$上的区间长度远大于$x$的下界。我们依赖于我们最近的工作最大值的$|ζ(1/2+it)|$长时间间隔的下界，以及工作的Soundararajan，Gál，和其他人。本文旨在展示和澄清概念上不同的组合参数，显示在不同的部分的证明。

Goss $t$ -adic Zeta函数的黎曼假设

Riemann Hypothesis for Goss $t$-adic Zeta Function

本文证明了高斯$v$ -adic zeta函数$ζ_{v}(s)$的黎曼假设，当$v$是一阶$\mathbb{F}_{q}[t]$的素数时.

A note on some infinite sums of Hurwitz zeta functions

We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $ζ(s,α)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $ζ(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.

黎曼ζ函数的临界零点

Critical zeros of the Riemann zeta-function

在这篇未发表的文章中，我们提出了一个想法，即使用一个三件式的软化器来稍微改善黎曼ζ函数在临界线上的零和单零点的已知百分比。这使用最近的结果Bettin，Bui，李和Radziwill [BBLR]扭曲的第四时刻的黎曼ζ函数。

Rational approximations to the zeta function II

This note describes continued fraction representations for the rational approximations to the zeta function recently found by the author. It is tempting to think that these continued fractions might be analysed using a souped up version of the Worpitzky argument so as to produce zero-free regions for the approximations.

On an analog of the Arakawa-Kaneko zeta function and relations of some 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.

Irrationalite d'au moins un des neuf nombres zeta（5），zeta（7），.，中文（简体）

Irrationalite d'au moins un des neuf nombres zeta(5),zeta(7),...,zeta(21)

我们证明了9个数zeta（5），zeta（7），.. zeta（21）。

Block shuffle identities for multiple zeta values

In 1998, Borwein, Bradley, Broadhurst and Lisoněk posed two families of conjectural identities among multiple zeta values, later generalized by Charlton using his alternating block notation. In this paper, we prove a new class of identities among multiple zeta values that simultaneously resolve and generalize these conjectures.