Convergence of zeta functions of graphs
Bryan Clair, Shahriar Mokhtari-Sharghi
无限图Y的$L^2$ -zeta函数（先前定义在围绕零的球中）具有解析扩张。对于一个由Y覆盖的有限图的塔，有限图的归一化zeta函数收敛于Y的$L^2$ -zeta函数。
The $L^2$-zeta function of an infinite graph Y (defined previously in a ball around zero) has an analytic extension. For a tower of finite graphs covered by Y, the normalized zeta functions of the finite graphs converge to the $L^2$-zeta function of Y.
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Some graph properties determined by edge zeta functions
Stark和Terras在1996年引入了有限图的边zeta函数。边缘zeta函数是多项式的倒数，其变量是图中边缘的两倍，并且可以在多项式时间内计算。我们看看我们可以使用边缘zeta函数确定的图形属性。特别地，边zeta函数足以推断团数、哈密顿圈的数目以及图是完美图还是弦图。实际上计算这些属性需要指数级的时间。最后，我们提出了一个新的例子，说明Ihara zeta函数不一定能做到这一点。
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from a vertex-transitive graph.
The theory of Ihara zeta functions is extended to infinite graphs which are weighted and of finite total weight. In this case one gets meromorphic instead of rational functions and the classical determinant formulas of Bass and Ihara hold true with Fredholm determinants.
Visible lattice points and the chromatic zeta function of a graph
我们研究了格[1，n]^s中长度为k的圈所包含的格点不多于圈的k个顶点的概率。然后，我们将这个问题推广到由给定图H导出的其他配置，引入了图的色zeta函数。
Zeta functions of graphs with $\mathbb{Z}$ actions
设$Y$是图$X$的一个正则覆盖，其覆盖变换群为$π= \mathbb{Z}$ .本文给出了$Y$的$L^2$ zeta函数的显式表达式，并给出了计算实例。当为$π= \mathbb{Z}$时，$L^2$ zeta函数为代数函数。作为结果，它扩展到一个亚纯函数上的黎曼曲面。亚纯扩张提供了一个设置，推广已知的正则图的zeta函数的性质，如奇点的位置和函数方程。
Spectral zeta functions of graphs and the Riemann zeta function in the critical strip
We initiate the study of spectral zeta functions $ζ_{X}$ for finite and infinite graphs $X$ , instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions . The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $ζ(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part o…
On the convergence of zeta functions of prehomogeneous vector spaces
本文证明了预齐次向量空间 zeta 函数的一般收敛性结果，推广了前人成果，分析了产生边界项的子空间，进一步在幂零轨道设定下研究，并确定了最右端极点处的留数。
The Zeta Function of a Hypergraph
我们以一种自然的方式将Ihara-Selberg zeta函数推广到超图。桥本的因子分解结果的双正则二部图适用，导致确切的因子分解。对于$(d,r)$ -正则超图，我们证明了一个修正的黎曼假设成立当且仅当超图是Winnie Li和Patrick Solé意义下的Ramanujan.最后，我们给出了一个例子来说明如何将广义zeta函数应用于图，以区分具有相同Ihara-Selberg zeta函数的非同构图。
The Ihara zeta function of the infinite grid
无限网格是具有通常生成元的$\mathbb{Z} \times \mathbb{Z}$的Cayley图。本文利用椭圆积分和θ函数计算了无限网格下的Ihara zeta函数。网格的zeta函数扩展为满足函数方程的解析多值函数。它的定义域中的奇点集是有限的。网格zeta函数是第一个计算的例子，它是非初等的，并且在其域的每个点上取无穷多个值。它也是正方形网格图和环面图的Ihara zeta函数的归一化序列的极限值。
We introduce a generalized Bartholdi zeta function for simple graphs with bounded degree. This zeta function is a generalization of both the Bartholdi zeta function which was introduced by L.~Bartholdi and the Ihara zeta function which was introduced by G.~Chinta, J.~Jorgenson and A.~Karlsson. Furthermore, we establish a Bartholdi type formula of this Bartholdi zeta function for simple graphs with bounded degree. Moreover, for regular graphs, we give a new expression of the heat kernel which is regarded as a one-parameter deformation of the expression obtained by G.~Chinta, J.~Jorgenson and A.…
A trace on fractal graphs and the Ihara zeta function
从1968年伊原的工作，有越来越多的兴趣研究zeta函数的有限图，由须田，桥本，巴斯，斯塔克和特拉斯，水野和佐藤，仅举几个作者。然后，克莱尔和Mokhtari-Sharghi研究了zeta函数的无限图作用于一个离散的自同构群。所有这些处理的主要公式建立了zeta函数（最初定义为无穷乘积）和图的拉普拉斯算子之间的联系。在这篇文章中，我们考虑另一类无限图。它们是分形图，即它们具有自相似性。我们定义了这些图的zeta函数，并使用算子代数的机器，我们证明了一个行列式公式，它涉及zeta函数与Laplacian的图。我们还证明了函数方程，和一个公式，它允许zeta函数的zeta函数的有限子图的近似。
Hurwitz-Lerch zeta and Hurwitz-Lerch type of Euler-Zagier double zeta distributions
本文利用Γ函数给出了具有$0 < σ\ne 1$的Hurwitz-Lerch zeta分布.此外，我们定义了Hurwitz-Lerch型的Euler-Zagier双zeta分布不仅在绝对收敛区域，而且在绝对收敛区域之外。
Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multizeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborifed zeta values are algebra morphisms for these new products on trees.
On convergence of points to limiting processes, with an application to zeta zeros
本文研究实轴上随机平移点序列的多种极限点过程收敛概念的等价性，证明相关Tauberian定理，并将结果应用于Riemann zeta函数零点，证明GUE假设的多种表述等价。
Let $K$ be a compact set with connected complement on the half-plane Re$(s)>0$, and let $f$ be a continuous function on $K$ which is analytic in its interior. We prove that for any parameter $0<α<1, α\neq \frac 1 2$ then $f(s)$ may be uniformly approximated arbitrarily closely by $ζ(1+iT+iδs,α)$ on $K$ for some $T,δ>0$, where $ζ(s,α)$ denote the Hurwitz zeta-function. This is the first known universality result that is also known to hold for the Hurwitz zeta-function with an algebraic irrational parameter.
On the Zeta functions of supersingular isogeny graphs and modular curves
本文证明了模曲线的 Hasse--Weil zeta 函数与配备级结构的超奇异椭圆曲线同源图的 Ihara zeta 函数之间的关系公式，推广了 Sugiyama 的结果。
We study some infinite products of absolute zeta functions. Especially, we consider the convergence and the rationality of them.
We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding…
Symmetric Schur multiple zeta functions
该研究引入具有对称函数结构的多重zeta函数，探究其基本性质，推导截断情形下的普法夫、行列式表达式与求和公式，并将结果推广至拟对称函数。
Ihara zeta functions for periodic simple graphs
本文综述了图的Ihara zeta函数的定义和主要性质，主要讨论了周期单图的情形。此外，我们给出了一个新的证明的相关行列式公式的基础上开发的处理由Stark和Terras有限图。
Zeta functions for infinite graphs and functional equations
综述了无限图的Ihara zeta函数和Bartholdi zeta函数的定义及其主要性质。讨论了函数方程的有效性的一般问题，并提出了各种可能的解决方案。
Special values of spectral zeta functions of graphs and Dirichlet L-functions
本文通过研究单位圆上一阶自伴微分算子的谱zeta函数，建立了一个组合导数公式，从而揭示了Dirichlet L-函数特殊值与圈图谱zeta函数特殊值之间的自然相互确定关系。
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。
Non-real poles on the axis of absolute convergence of the zeta functions associated to Pascal's triangle modulo a prime
从Pascal三角形中选取不能被给定素数整除的二项式系数，我们发现它们形成一个具有自相似性的集合。Essouabri研究了与上述集合相关的一类亚纯函数。这些函数与分形几何有关，并且这样的函数在其绝对收敛轴上是否具有非实极点是一个问题。Essouabri给出了一个证明存在这样一个非真正的极在最简单的情况下。他的证明的关键是斯坦因和威尔逊对帕斯卡三角形模一个素数中的点相乘的速度的估计。本文将Essouabri的结果推广到某些情形，并给出了不同于传统的Pascal模素数三角形点数的计算方法。
We give explicit expressions (or at least an algorithm of obtaining such expressions) of the coefficients of the Laurent series expansions of the Euler-Zagier multiple zeta-functions at any integer points. The main tools are the Mellin-Barnes integral formula and the harmonic product formulas. The Mellin-Barnes integral formula is used in the induction process on the number of variables, and the harmonic product formula is used to show that the Laurent series expansion outside the domain of convergence can be obtained from that inside the domain of convergence.
Hurwitz zeta and Euler-Zagier-Hurwitz type of double zeta distributions and real zeros of these zeta functions
本文利用Γ函数给出了具有$0 < σ\ne 1$的Hurwitz zeta分布.在证明过程中，我们证明了Hurwitz zeta函数$ζ(σ,a)$对所有$0 <σ<1$不为零当且仅当$a \ge 1/2$ .其次，我们定义了Euler-Zagier-Hurwitz型双zeta分布，不仅在绝对收敛区域，而且在绝对收敛区域之外。证明了双zeta函数$ζ_2 (σ_1,σ_2\,;a)$的Euler-Zagier-Hurwitz型在$0<σ_1<1$，$σ_2>1$和$1<σ_1+σ_2<2$时不为零当且仅当$a \ge 1/2$为零.
Zeta Functions of Complexes Arising from PGL(3)
本文得到了非阿基米德局部域$F$上$PGL_3$的Bruhat-Tits构作的有限商$X_Γ= Γ\backslash PGL_3(F)/PGL_3(O_F)$的zeta函数$Z(X_Γ, u)$的封闭形式表达式.类似于图zeta函数，$Z(X_Γ, u)$是有理函数，并且它满足黎曼假设当且仅当$X_Γ$是Ramanujan复形。
Kontsevich graphs associated with specific multiple zeta values
本文显式构造了Kontsevich图，其积分给出特定多重zeta值，并确定了产生给定权重的归一化多重zeta值的图组合的权重。
The convergence of a sequence of Cauchy sequences is conjectured; which if shown to be true, would prove the Riemann hypothesis by way of LeClair and França's transcendental equation criteria.
A functional relation for Tornheim's double zeta functions
本文推广了重zeta值理论中的部分分式分解，证明了三个复变数的Tornheim双zeta函数之间的一个关系.作为应用，我们给出了几个zeta函数的新的整表示，将宇称结果推广到整个收敛域，给出了Tornheim双zeta函数在非正整数处的具体表示，以及某个Witten zeta函数在每一整数处的一些性质.作为附录，我们给出了欧拉双zeta函数的一个函数方程。
Well-rounded zeta-function of planar arithmetic lattices
本文研究了平面上固定算术格的良圆子格的zeta函数的性质。特别地，我们证明了该函数在实极点为2阶的$s=1$处具有收敛横坐标，改进了S.库恩莱因我们利用这个结果证明了指数小于或等于$N$的平面算术格的圆满子格的个数是$O(N \log N)$ as $N \to \infty$。为了得到这些结果，我们产生了一个完整的完整的圆润的子格的一个固定的平面完整的圆润格的描述，并调查这种格的相似类的zeta函数的收敛性质，建立在作者以前的一些结果。
Recovering Laplacian Lattices from $L$-Functions of Graphs
该研究将代数曲线的L-函数推广到图论语境，定义了新的图L-函数，证明无桥图的雅可比行列式与该L-函数可唯一确定拉普拉斯格，同时指出洛伦齐尼zeta函数在桥收缩下不变，且仅雅可比行列式或洛伦齐尼zeta函数无法单独确定拉普拉斯格。
Convergence of random holomorphic functions with real zeros and extensions of the stochastic zeta function
本文建立收敛点过程到零点由其确定的随机全纯函数泛函收敛的统一框架，推广随机矩阵特征多项式与随机 zeta 函数结果，并证明极限点过程的 Stieltjes 变换几乎处处服从标准 Cauchy 分布。
This paper discusses the simplest examples of spectral zeta functions, especially those associated with graphs, a subject which has not been much studied. The analogy and the similar structure of these functions, such as their parallel definition in terms of the heat kernel and their functional equations, are emphasized. Another theme is to point out various contexts in which these non-classical zeta functions appear. This includes Eisenstein series, the Langlands program, Verlinde formulas, Riemann hypotheses, Catalan numbers, Dedekind sums, and hypergeometric functions. Several open-ended pr…
The expansion of a modular graph function on a torus of modulus $τ$ near the cusp is given by a Laurent polynomial in $y= π\Im (τ)$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term whose coefficient is rational and exponentially suppressed terms. We prove that the coefficients of the non-leading terms in the Laurent polynomial of the modular graph function $D_N(τ)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary…
We revisit the theory of Ihara $L$-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study if the Artin formalism is satisfied by these $L$-functions. As an application, we give a proof of the analogue of Iwasawa's asymptotic class number formula for the $p$-part of the number of spanning trees in branched $\mathbb{Z}_{p}$-towers of finite connected graphs using Ihara zeta and $L$-functions. Moreover, we relate a generator for the characteristic ideal of the finitely generated torsion Iwasawa module that governs the growth of the …
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
本文提出一种加速振荡级数收敛的算法，并将其应用于多对数函数和Hurwitz zeta函数的高效计算，扩展了Borwein的Riemann zeta函数算法。
Zeta functions, heat kernels and spectral asymptotics on degenerating families of discrete tori
离散环面是指与有限循环群的有限乘积相关联的凯莱图，其生成集通过为每个循环因子选择生成元来给出。在这篇文章中，我们研究了离散环面序列的组合拉普拉斯算子的谱理论，当循环因子的阶以可比的速率趋于无穷大时。首先，我们表明，热内核对应于退化家庭的序列收敛，重新缩放后，在一个相关的真正的环面上的热内核。然后，我们建立了一个渐近展开，在退化参数，组合拉普拉斯行列式。极限实环面的拉普拉斯算子的ζ正则化行列式在这种展开式中表现为常数项。另一方面，使用基尔霍夫的经典定理，有限图的组合拉普拉斯算子的行列式除以顶点数等于图的生成树的数量，称为复杂度。结果，我们建立了有限阿贝尔群的Cayley图的复杂性与实环面高度之间的精确联系。还已知离散环面上的谱行列式可以用三角函数表示，而实环面上的谱行列式可以用一般线性群上的模形式表示。因此，我们的分析的另一种解释是在三角函数和模形式的某些乘积的极限值之间建立联系。热核分析，我们采用了仔细研究的I-Bessel函数。我们的方法扩展到通过退化证明其他谱不变量的渐近行为，如谱zeta函数和Epstein-Hurwitz型zeta函数的特殊值。
We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series in hypergeometric functions on each side. These equalities capture basic analytic properties of the Hurwitz zeta function like meromorphic extension and the functional equation. We give rates of convergence for the involved series. As a special case we find that the exponential rate of convergence for the deformation of the log-sine to the fractional part in…
The distribution of the logarithmic derivative of the Riemann zeta-function
本文研究了黎曼ζ函数的对数导数在Re（s）=σ线上的分布，其中σ在临界线σ=1/2附近的一定范围内.对于这样的σ，我们证明了在复平面上，σ ′/σ（s）的分布收敛于一个二维高斯分布.上界的收敛速率的高斯分布也得到了。
In this PhD thesis we study holomorphic and non-holomorphic elliptic analogues of multiple zeta values, namely elliptic multiple zeta values and modular graph functions. Both classes of functions have been discovered very recently, and are involved in the computation of genus one superstring amplitudes. In particular, we obtain new results on the asymptotic expansion of these functions that allow us to perform explicit computations and point out analogies between genus zero and genus one amplitudes.
We relate one-loop scattering amplitudes of massless open- and closed-string states at the level of their low-energy expansion. The modular graph functions resulting from integration over closed-string punctures are observed to follow from symmetrized open-string integrals through a tentative generalization of the single-valued projection known from genus zero.
Zeros of the alternating zeta function on the line R(s)=1
交替zeta函数zeta*（s）= 1 - 2^{-s} + 3^{-s} -.通过恒等式（1-2^{1-s}）zeta（s）= zeta*（s）与黎曼ζ函数相关。我们推导出zeta*（s）在因子1-2^{1-s}的每个非实零点处的消失，而不使用恒等式。相反，我们使用一个公式将zeta*（s）级数的部分和与x^{-s}从x=1到x=2的积分的黎曼和联系起来。我们将证明与我们早期的论文“黎曼假设，单零点，以及不当黎曼和的渐近收敛度”（Proc. Amer. Math. Soc. 126（1998）1311-1314）联系起来。
On A Rapidly Converging Series For The Riemann Zeta Function
黎曼zeta函数的计算对于数论领域的许多研究都是很重要的，特别是对于快速收敛的级数。本文研究了一类求和公式，作为特例，给出了黎曼ζ函数快速收敛级数的一个新证明。该系列收敛于整个复平面，其收敛速率明显快于可比的表示，因此是一个有用的基础评估算法。相应系数的评估是没有问题的，精确的收敛速度详细阐述。得到的全局收敛序列允许减少黎曼假设的多项式类似的属性。有趣的是，拉盖尔的多项式形成了贯穿所有章节的主旋律。
Zeta functions of abstract isogeny graphs and modular curves
本文提出抽象同源图这一 Serre 图的不可定向变体，证明其 Ihara 行列式公式类比，并将超奇异椭圆曲线同源图的 Ihara Zeta 函数与模曲线的 Hasse-Weil Zeta 函数联系起来，进而给出点计数与圈计数的显式关系及非回溯圈数的渐近公式。
Graph integrals, Feynman periods, and single-valued multiple zeta values
研究图积分是什么数的问题，通过证明原始规范积分与变形量子化中的复位置空间积分一致，得出图的规范积分值为单值多重zeta值，且每个单值多重zeta值是特定图费曼周期的有理线性组合，还表明在交换图复形中相关上闭链一致。
On the series expansion of the secondary zeta function about $s=1$ and its coefficients
本文研究二次ζ函数在s=1处的展开系数，推导新公式并验证其精度，同时应用Brent定理提升收敛性。
Opposite Sign Kloosterman Sum Zeta Function
研究了符号相反的Kloosterman和zeta函数$$(2π\sqrt{mn})^{2s-1}\sum_{\ell=1}^\infty \frac{S(m,-n,\ell)}{\ell^{2s}}$$（$m,n$为正整数）到所有$s \in \mathbb{C}$的亚纯延拓和谱展开.该函数的极点对应于黎曼ζ函数的零点和Maass形式的谱参数。这个函数的解析性质相当微妙。证明了zeta函数的谱展开式仅在左半平面内收敛，与狄利克雷级数的绝对收敛区域不相交，即使它们都是整个复平面上同一亚纯函数的解析表达式。
We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically, we study the family of functions $V_z: s \mapsto ζ(s) \exp (zs)$. We observe convergence of $V_z$ fixed points along nearly logarithmic spirals with initial points at zeta fixed points and centered upon Riemann zeros. We can approximate these spirals numerically, so they might afford a means to study the geometry of the relationship of zeta fixed points to Riemann zeros.
We construct a new class of directed and bipartite random graphs whose topology is governed by the analytic properties of L-functions. The bipartite L-graphs and the multiplicative zeta graphs are relevant examples of the proposed construction. Phase transitions and percolation thresholds for our models are determined.
Evaluation of Riemann Zeta function on the Line $\Re(s) = 1$ and Odd Arguments
本文针对黎曼ζ函数在奇数参数和直线$\Re(s) = 1$上的评估，提出了一个具有指数收敛性的简单近似公式，并与文献中的其他公式进行了比较。
Analytic properties of representation zeta functions of groups of type $\mathsf{A_2}$
本文研究了诸如 $\textrm{SL}_3(\mathbb{Z})$ 等 $\mathsf{A}_2$ 型算术群的表示 zeta 函数的解析性质，发现了更多极点并确定了亚纯延拓的自然边界，同时分析了数域和函数域两种情况。
A Variant of the Fractional Gamma-regularized Zeta Kernel
本文引入分数伽马正则化泽塔核的变体，研究其解析性质与收敛性，证明其作为黎曼泽塔函数的形变，并在特定极限下插值到交替泽塔函数，同时提供狄利克雷级数表示。
On the relation between quantum walks and absolute zeta functions
本文建立图上 Grover walk 与绝对 zeta 函数的联系，证明量子游走诱导的 zeta 函数构成依赖边数的绝对自守形式，并以循环图为例将其绝对 zeta 函数表示为二阶多重 gamma 函数。
We prove convergence of a stochastic approximation of powers of the Riemann $ζ$ function to a non-Gaussian multiplicative chaos measure, and prove that this measure is a non-trivial multifractal random measure. The results cover both the subcritical and critical chaos. A basic ingredient of the proof is a 'good' Gaussian approximation of the induced random fields that is potentially of independent interest.
Orbits for products of maps
研究了映射乘积的动态zeta函数和轨道狄利克雷级数的性质。讨论了Zeta函数的收敛半径收敛半径与轨道狄利克雷级数的收敛横坐标乘积下的性质。本文证明了每种长度都有一条轨道的映射的立方的轨道狄利克雷级数有自然边界。
We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane. As an application, we show that ideal zeta functions associated with nilpotent Lie algebras of maximal class have abscissa of convergence $2$.
With this paper we introduce a new series representation of $ζ(3)$, which is based on the Clausen representation of odd integer zeta values. Although, relatively fast converging series based on the Clausen representation exist for $ζ(3)$, their convergence behavior is very slow compared to BBP-type formulas, and as a consequence they are not used for explicit numerical computations. The reason is found in the fact that the corresponding Clausen function can be calculated analytically for a few rational arguments only, where $x=\frac{1}{6}$ is the smallest one. Using polylogarithmic identities …
Mixed zeta functions and application to some lattice points problems
我们考虑zeta函数：$Z(f ;P ;s)=\sum_{\m \in \N^{n}} f(m_1,..., m_n) P(m_1,..., m_n)^{-s/d}$，其中$P \in \R [X_1,..., X_n]$具有次数$d$，并且$f$是原始算术函数，例如乘性函数。本文研究了当$f$和$P$满足应用中通常遇到的性质时，这类级数在先验绝对收敛域之外的亚纯延拓.因此，我证明了一个明确的渐近的一般类格点问题的算术约束。
In the present series of papers, we study the behavior of the r-fold zeta-function of Euler-Zagier type with identical arguments on the real line. In this first part, we consider the behavior on the interval [0,1]. Our basic tool is an "infinite" version of Newton's classical identities. We carry out numerical computations, and draw graphs for real s in [0,1], for several small values of r. Those graphs suggest various properties of the r-fold zeta-function, some of which we prove rigorously. For example, we show that the r-fold zeta-function has r asymptotes, and determine the asymptotic beha…
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