On Witten multiple zeta-functions associated with semisimple Lie algebras V

Witten多重zeta值附加到sl（4）

Witten multiple zeta values attached to sl(4)

本文证明了，除9种非正则情形外，sl（4）上每个权w>3的Witten重zeta值都是权w和深度小于3的MZVs的有限有理线性组合，其中还需要Riemann zeta值zeta（w-2）和权w-1和深度<3的双zeta值。

On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function

We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number of n-dimensional representations of SU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.

Witten zeta函数在负整数处的消失

Vanishing of Witten zeta function at negative integers

利用涉及Hurwitz zeta函数的积分表示，证明了根系Φ的Witten zeta函数在负偶数处具有高阶消失，解决了Kurokawa和Ochiai对一大类紧李群的猜想，并给出了相应首项系数的定性描述。

SO（5）上Witten重zeta函数的交错Euler和与特殊值

Alternating Euler sums and special values of Witten multiple zeta function attached to so(5)

本文研究Matsumoto定义的李代数so（5）的Witten重zeta函数。我们的主要结果表明，它在非负整数上的特殊值总是可以用交错欧拉和来表示。更准确地说，每个这样的特殊权重值w>2是权重w和深度至多为2的交替欧拉和的有限有理线性组合，除了当唯一的非零自变量是最后两个变量之一时，在这种情况下需要$ζ(w-1)$。

关于秩2与秩3的单变量Witten zeta函数

On single-variable Witten zeta functions of rank two and three

该研究通过引入Mellin变换的新型积分核，探究秩2与秩3的单变量Witten zeta函数性质，得到极点、留数等相关结果，提出原点导数猜想，并讨论其与Eisenstein级数的关联及p进性质。

超平面排列的组合学与原点处的Witten zeta函数

Combinatorics of hyperplane arrangements and Witten zeta function at the origin

本文提出一种新方法，利用超平面排列的组合学，为Witten zeta函数在原点的值及其导数导出优雅公式，并推广至锥形zeta函数族。

Functional relations for zeta-functions of weight lattices of Lie groups of type $A_3$

We study zeta-functions of weight lattices of compact connected semisimple Lie groups of type $A_3$. Actually we consider zeta-functions of SU(4), SO(6) and PU(4), and give some functional relations and new classes of evaluation formulas for them.

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.

Functional equations for Weng's zeta functions for $(G,P)/\mathbb{Q}$

It is shown that Weng's zeta functions associated with arbitrary semisimple algebraic groups defined over the rational number field and their maximal parabolic subgroups satisfy the functional equations.

A functional relation for Tornheim's double zeta functions

In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give new integral representations of several zeta functions, an extension of the parity result to the whole domain of convergence, concrete expressions of Tornheim's double zeta function at non-positive integers and some results for the behavior of a certain Witten's zeta function at each integer. As an appendix, we show a functional equation for Euler's double ze…

Expressions of Schur multiple zeta-functions of anti-hook type by zeta-functions of root systems

We investigate relations among Schur multiple zeta functions and zeta-functions of root systems attached to semisimple Lie algebras. Schur multiple zeta functions are defined as sums over semi-standard Young tableaux. Then, assuming the Young tableaux is of anti-hook shape, we show that they can be written in terms of modified zeta-functions of root systems of type $A$. Our proof is quite computational, but we also give a pictorial interpretation of our argument in terms of Young tableaux. It is also possible to understand that one of our theorems gives an expression of Schur multiple zeta fun…

维滕zeta函数的一个通用首项留数公式

A universal leading-residue formula for Witten zeta functions

研究维滕zeta函数，通过将首项格系数与临界指数处积分识别并利用相关恒等式计算，证明其在$s = 2/h$有简单极点及留数公式，确立猜想并获渐近式，给出维数至多为$X$的不可约表示数量的非陶伯型渐近。

经典威滕ζ函数的移位极点与腔抵消

Shifted poles and chamber cancellation for classical Witten zeta functions

该研究确定经典单变量威滕ζ函数正实轴上的两类无限极点族，推导相关留数与首项系数表达式，结合多种方法完成证明，未用数值非零估计。

非阿基米德局部域上有限维半单代数中序上格的所罗门zeta函数

Solomon zeta functions over arithmetic orders

研究非阿基米德局部域上有限维半单代数中序上格的所罗门zeta函数，通过代数方法将部分所罗门zeta函数与极大序zeta函数的商表示为有限和，得到相关显式公式，包括\(\mathbb{Z}_p[\mathbb{Z}/p\mathbb{Z}]\)上非投射格的公式。

Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line

In this paper, we show that any polynomial of zeta or $L$-functions with some conditions has infinitely many complex zeros off the critical line. This general result has abundant applications. By using the main result, we prove that the zeta-functions associated to symmetric matrices treated by Ibukiyama and Saito, certain spectral zeta-functions and the Euler-Zagier multiple zeta-functions have infinitely many complex zeros off the critical line. Moreover, we show that the Lindelöf hypothesis for the Riemann zeta-function is equivalent to the Lindelöf hypothesis for zeta-functions mentioned a…

Representation growth and representation zeta functions of groups

We give a short introduction to the subject of representation growth and representation zeta functions of groups, omitting all proofs. Our focus is on results which are relevant to the study of arithmetic groups in semisimple algebraic groups, such as the special linear group of degree n over the ring of integers. In the last two sections we state several results which were recently obtained in joint work with N. Avni, U. Onn and C. Voll.

高斯超几何方程解的多重多项式表示、多重多项式的函数关系和多重zeta值的关系

Representation of solutions of the Gauss hypergeometric equation by the multiple polylogarithms, functional relations of the multiple polylogarithms and relations of the multiple zeta values

本文利用迭代积分将高斯超几何方程的解表示为多重多项式的级数。这种表示是形式KZ方程解的半简单表示的最简单的情况。此外，将该表示与Gauss超几何方程解的联系关系相结合，得到了一元多重多项式与多重zeta值的各种关系.

广义多重 Hurwitz zeta 函数在非正整数处的值与导数值

Values and derivative values at nonpositive integers of generalized multiple Hurwitz zeta functions

本文建立广义 Hurwitz 型多重 zeta 函数的亚纯延拓，导出其沿给定方向在非正整数处的值及导数值公式，并应用于 Witten zeta 函数与 $\mathfrak{g}_2$ 表示数的渐近分析。

A study on multiple zeta values from the viewpoint of zeta-functions of root systems

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 …

Asymptotic expansions for partitions generated by infinite products

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $\mathcal{L}$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree $n$ of $\mathfrak{so}{(5)}$. We also study the Witten zeta function $ζ_{\mathfrak{so…

General Uniformity of Zeta Functions

Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the singularities of analytic torsions at Brill-Noether loci, and the asymptotic behaviors of analytic torsions with respect to the degree. These new yet intrinsic zetas, both abelian and non-abelian, are expected to play key roles to understand global analysis and geometry of Riemann surfaces, such as the Tamagawa number conjecture for Riemann surfaces, searched …

On the geometric side of the Arthur trace formula for the symplectic group of rank 2

We study the non-semisimple terms in the geometric side of the Arthur trace formula for the split symplectic similitude group or the split symplectic group of rank 2 over any algebraic number field. In particular, we show that the coefficients of unipotent orbital integrals are expressed by the Dedekind zeta function, Hecke L-functions, and the Shintani zeta function for the space of binary quadratic forms.

L函数的Dedekind和、Eisenstein上循环及特殊值的计算

Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions

我们定义了某些高维的Dedekind和，推广了经典的Dedekind-Rademacher和，并展示了如何使用推广的连分式算法有效地计算它们。我们提出了两个应用程序。首先，我们展示了如何通过第二作者引入的Eisenstein上循环来表示与全实数域相关的偏zeta函数的特殊值。因此，我们得到一个多项式时间算法计算这些特殊的值。其次，我们展示了如何使用我们的技术来计算某些特殊值的Witten zeta函数，并计算一些明确的例子。

Dirichlet物种与算术ζ函数

Dirichlet Species and Arithmetic Zeta Functions

本文提出通过物种理论范畴化算术ζ函数，利用有限集结构生成半单交换环并编码有限域上的点数。

The representation zeta function of a FAb compact p-adic Lie group vanishes at -2

Let G by compact p-adic Lie group and suppose that G is FAb, i.e., that H/[H,H] is finite for every open subgroup H of G. The representation zeta function Z(G,s) encodes the distribution of continuous irreducible complex characters of G. Here s denotes a complex variable and Z(G,s) is defined as the Dirichlet generating function whose nth coefficient is equal to the number of irreducible characters of G of degree n. For p greater than 2 it is known that Z(G,s) defines a meromorphic function on the complex plane. Wedderburn's structure theorem for semisimple algebras implies that ZG,-2) = |G| f…

3-流形的$L$-函数不变量与广义伯努利多项式之间的关系

$L$-function invariants for 3-manifolds and relations between generalized Bernoulli polynomials

该研究定义了负定型配管流形的$L$-函数，证明其为整函数且$s=0$处值对应Witten--Reshetikhin--Turaev不变量，还得到了这类推广型zeta函数负整数特殊值的线性关系。

与李代数相关的多项式

Polynomials associated to Lie algebras

本文研究了与李代数相关的多项式，通过Wittenζ函数的特殊值定义，并与2008年Komori等人定义的多项式进行比较。

关于theta函数的比值

On ratios of theta functions

本文研究theta函数比值的极值问题，发现正六边形晶格在其中起关键作用，并应用于共形场论和李乌ville场论，同时对晶体学和相互作用粒子理论有影响。

凸域的热带zeta函数的留数

Residues of a tropical zeta function for convex domains

定义凸域的SL_n(Z)不变热带zeta函数，证明其在C^3严格凸域上可亚纯延拓至Re(s)>3/5，在s=2/3处有单极点且留数与等仿射周长成正比，并得到波前晶格周长渐近式。

代数群自同态的动力系统

Dynamics of endomorphisms of algebraic groups

本文推广 Steinberg 不动点公式至任意光滑代数群，引入 FAD-序列公理化不动点序列，证明 Artin-Mazur zeta 函数二分法并研究周期轨道计数的素数定理类比。

纤维化Gromov-Witten理论、旗丛的量子谱与整数的素因子分解

Fiberwise Gromov-Witten theory, quantum spectra of flag bundles, and prime factorization of integers

本文研究旗丛的垂直量子上同调与量子谱，揭示其退化由素因子分解控制，并引入双序列族，其解析性质联系黎曼zeta函数与哥德巴赫猜想。

六胶子开超弦盘振幅、多重超几何级数与Euler-Zagier和

Six Gluon Open Superstring Disk Amplitude, Multiple Hypergeometric Series and Euler-Zagier Sums

用超弦理论计算了六胶子盘的振幅。这个振幅探测了规范与DP膜上六个外部支腿的相互作用，特别是包括例如F^6项。全弦S-矩阵可以用六个广义多重超几何函数（三重超几何函数）表示，它们在有效作用量中对安排高阶α '规范相互作用项（如F^6，D^4 F^4，D^2 F^5，D^6 F^4，D^2 F^6，.）一个系统的和有效的方法被发现来计算树级弦振幅相等的看似不同的表达式为一个和相同的字符串S-矩阵：比较出现在弦环计算的黎曼恒等式，我们发现一个有趣的方式使用世界片超对称性，以产生一个系统的非平凡方程的弦树级振幅。这些方程导致不同的多重超几何函数之间的代数恒等式。他们的（六维）解给出了弦S-矩阵的成分。我们导出与任何开弦六点散射过程相关的材料：三重超几何函数、它们的积分表示和它们由（广义）欧拉-扎吉埃和或（相关）威滕ζ函数给出的α '（动量）展开之间的关系。