Zeta functions do not determine class numbers

Determination of the real poles of the Igusa zeta function for curves

The numerical data of an embedded resolution determine the candidate poles of Igusa's p-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case.

On mixed joint discrete universality for a class of zeta-functions III

We present the most general at this moment results on the discrete mixed joint value-distribution and the universality property for the class of Matsumoto zeta-functions and periodic Hurwitz zeta-functions under certain linear independence condition on the relevant parameters, such as common differences of arithmetic progressions, prime numbers etc.

A modification of the mixed joint universality theorem for a class of zeta-functions

The property of zeta-functions on mixed joint universality in the Voronin's sense states that any two holomorphic functions can be approximated simultaneously with accuracy $ε>0$ by suitable vertical shifts of the pair consisting from the Riemann zeta- and Hurwitz zeta-functions. In [1], it was shown rather general result, i.e., an approximating pair was composed of the Matsumoto zeta-functions' class and the periodic Hurwitz zeta-function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably many $ε>0$. Also, we give the concluding remark…

Remarks on the mixed joint universality for a class of zeta-functions

Two remarks related with the mixed joint universality for a polynomial Euler product and a periodic Hurwitz zeta-function with a transcendental parameter are given. One is the mixed joint functional independence, and the other is a generalized universality, which includes several periodic Hurwitz zeta-functions.

关于预齐次向量空间 zeta 函数的收敛性

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。

图的谱zeta函数特殊值与Dirichlet L-函数

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。

The discrete case of the mixed joint universality for a class of certain partial zeta-functions

We give a new type of mixed discrete joint universality properties, which is satisfied by a wide class of zeta-functions. We study the universality for a certain modification of a Matsumoto zeta-function and a periodic Hurwitz zeta-function under certain arithmetic condition on the common difference of arithmetical progression and the shifting parameter. Also we generalize a known discrete universality result in our former paper and the new one as well.

Ihara zeta functions and class numbers

The prime geodesic theorem for cycles in Bruhat-Tits buildings is applied to unit groups of division algebras to derive new asymptotic assertion on class numbers of orders in imaginary quadratic fields.

Regularized product expressions of higher Riemann zeta functions

As a generalization of [KMW], we introduce a higher Riemann zeta function for an abstract sequence. Then we explicitly determine its regularized product expression.

Igusa local zeta functions of a class of hybrid polynomials

In this paper, we study the Igusa's local zeta functions of a class of hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. Such class of hybrid polynomial was first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We prove the rationality of these local zeta functions and describe explicitly their poles. The proof is based on Igusa's stationary phase formula.

Depth reduction of a class of Witten zeta functions

We show that if a,b,c,d,f are positive integers such that a+b+c+d+f is even, then the Witten zeta value zeta_{sl(4)}(a,b,c,d,0,f) is expressible in terms of Witten zeta functions with fewer arguments.

黎曼 zeta 函数零点的显式公式

An explicit formula for the zeros of the Riemann zeta function

本文提出一个用素数及 Hermite 多项式加权和表示黎曼 zeta 函数零点加权和的显式公式，证明素数决定零点分布，并可在不使用 zeta 函数本身的情况下计算其零点。

Multiple zeta-star values and multiple integrals

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.

Hypergeometric Zeta Functions

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

On mixed joint discrete universality for a class of zeta-functions: One more case

We prove a new case of mixed discrete joint universality theorem on approximation of certain target couple of analytic functions by the shifts of a pair consisting of the function belonging to wide class of Matsumoto zeta-functions and the periodic Hurwitz zeta-function. We work under the condition that the common difference of arithmetical progression satisfies a certain rational condition and the parameter is a transcendental number. The essential difference from the result in our previous article is that here we do not study the class of partial zeta-functions, but work with the class of th…

Stability results for local zeta functions of groups and related structures

Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of these operations can be expressed simultaneously in terms of a single explicit formula. We show that in these cases, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions,…

Zeros of the extended Selberg class zeta-functions and of their derivatives

Levinson and Montgomery proved that the Riemann zeta-function $ζ(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $ζ'(1/2+it)=0$ implies $ζ(1/2+it)=0$. Here we obtain that in small areas located to the left of the critical line and near it the functions $ζ(s)$ and $ζ'(s)$ have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class $S$. We also consider zero trajectories of a certain family of zeta-functions from $S$.

Values of symmetric polynomials and a truncated analogue of the Riemann zeta function

For each positive integer n, we determine the set of symmetric functions f for which the congruence f(p/1,p/2,...,p/(p-1)) \equiv 0 mod p^n holds for all sufficiently large primes p. Our determination is conditional on a conjecture regarding the modulo p independence of Bernoulli numbers. In a recent work the author introduced a new truncated analogue of the multiple zeta function and investigated a class of relations among values of this function at positive integers. The question answered in the present work is equivalent to the determination of the relations satisfied by values of the corre…

A class of relations among multiple zeta values

We prove a new class of relations among multiple zeta values (MZV's) which contains Ohno's relation. We also give the formula for the maximal number of independent MZV's of fixed weight, under our new relations. To derive our formula for MZV's we consider the Newton series whose values at non-negative integers are finite multiple harmonic sums.

Tree-lattice zeta functions and class numbers

The theory of Ihara zeta functions is extended to non-compact arithmetic quotients of Bruhat-Tits trees. This new zeta function turns out to be a rational function, despite the infinite-dimensional setting. In general it has zeros and poles, in contrast to the compact case. The determinant formulas of Bass and Ihara hold true if one defines the determinant as limit of all finite principal minors. From this analysis, a prime geodesic theorem is derived, which, applied to special arithmetic groups, yields new asymptotic assertions on class numbers of orders in global fields.

Partial zeta functions, partial exponential sums, and p-adic estimates

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, …

Empirically determined Apery-like formulae for zeta(4n+3)

Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and appears to give the best possible series relations of this type, at least for n<12. The formula reduces to a finite but apparently non-trivial combinatorial identity. The identity is equivalent to an interesting new integral evaluation for the central binomial coefficient. We outline a new technique for transforming and summing certain infinite series. We also derive a beautiful formula which provide…

$\mathsf{A_2}$ 型群的表示 zeta 函数的解析性质

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。

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。

On mixed joint discrete universality for a class of zeta-functions II

We study analytic properties of the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with a transcendental parameter. We first survey briefly previous results, and then investigate the mixed joint discrete value distribution and prove the mixed joint discrete universality theorem for these functions, in the case when common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are necessary.

Hybrid moments of the Riemann zeta-function

The "hybrid" moments $$ \int_T^{2T}|ζ(1/2+it)|^k{(\int_{t-G}^{t+G}|ζ(1/2+ix)|^\ell dx)}^m dt $$ of the Riemann zeta-function $ζ(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_ε(T^{1+ε}G^m)$. This is shown to be true for certain specific values of the natural numbers $k,\ell,m$, and the explicitly determined range of $G = G(T;k,\ell,m)$. The application to a mean square bound for the Mellin transform function of $|ζ(1/2+ix)|^4$ is given.

Moment Zeta Functions for Toric Calabi-Yau Hypersurfaces

We study in detail the family of Calabi-Yau hypersurfaces defined by x_1+...+x_n+1/(x_1...x_n)=t over a finite field k. We determine its local and global monodromy and the trivial factors of its moment zeta function and Dwork's unit root zeta function.

On mixed joint discrete universality for a class of zeta-functions

We prove a mixed joint discrete universality theorem for a Matsumoto zeta-function $φ(s)$ (belonging to the Steuding subclass) and a periodic Hurwitz zeta-function $ζ(s,α;{\mathfrak{B}})$. For this purpose, certain independence condition for the parameter $α$ and the minimal step of discrete shifts of these functions is assumed. This paper is a continuation of authors' works [12] and [arXiv:1601.03795].

The Lerch zeta function IV. Hecke operators

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned…

Structural properties of multiple zeta values

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.

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.

Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two bridge link complements

Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two bridge link groups are determined. Special values of the zeta functions at s=0,1,2 are also investigated.

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.

Hurwitz zeta函数的四阶矩

The fourth moment of the Hurwitz zeta function

该研究针对平移参数为无理性指数小于3的无理数的Hurwitz zeta函数，给出其临界线上四阶矩的精确上界，确定了0≤k≤2时2k阶矩的量级，推测其临界线上值分布为高斯分布。

Goldbach表示的平均数与Riemann zeta-函数的无零点区域

The Average Number of Goldbach Representations and Zero-Free Regions of the Riemann Zeta-Function

本文证明了Goldbach表示平均数的Fujii公式的无条件形式，揭示了公式误差与Riemann zeta-函数无零点区域之间的相互决定关系，并通过素数定理余项将误差与无零点区域联系起来。

Associating curves of low genus to infinite nilpotent groups via the zeta function

It is known from work of du Sautoy and Grunewald in \cite{duSG1} that the zeta functions counting subgroups of finite index in infinite nilpotent groups depend upon the behaviour of some associated system of algebraic varieties on reduction $\modp.$ Further to this, in \cite{duS3, duS4} du Sautoy constructed a group whose local zeta function was determined by the number of points on the elliptic curve $E:Y^2=X^3-X.$ In this work we generalise du Sautoy\rq s construction to define a class of groups whose local zeta functions are dependent upon the number of points on the reduction of a given el…

On the Northcott property of zeta functions over function fields

Pazuki and Pengo defined a Northcott property for special values of zeta functions of number fields and certain motivic $L$-functions. We determine the values for which the Northcott property holds over function fields with constant field $\mathbb{F}_q$ outside the critical strip. We then use a case by case approach for some values inside the critical strip, notably $Re (s) < \frac{1}{2} - \frac{\log 2}{\log q}$ and for $s$ real such that $1/2 \leq s \leq 1$, and we obtain a partial result for complex $s$ in the case $1/2< Re(s)\leq 1$ using recent advances on the Shifted Moments Conjecture ov…

The Lerch Zeta Function I. Zeta Integrals

This is the first of four papers that study algebraic and analytic structures associated to the Lerch zeta function. This paper studies "zeta integrals" associated to the Lerch zeta function using test functions, and obtains functional equations for them. Special cases include a pair of symmetrized four-term functional equations for combinations of Lerch zeta functions, found by A. Weil, for real parameters (a,c) with 0< a, c< 1. It extends these functions to real a, and c, and studies limiting cases of these functions where at least one of a and c takes the values 0 or 1. A main feature is th…

Subconvexity of twisted Shintani zeta functions

Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version.

The Values of the Riemann Zeta-Function on Discrete Sets

We study the values taken by the Riemann zeta-function $ζ$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $ζ$ taken on this set. Moreover, we prove a joint discrete universality theorem for $ζ$ with respect to certain permutations of the set of positive integers. Finally, we study a generalization of the classical denseness theorems for $ζ$.

Zeta functions of trinomial curves and maximal curves

We determine the zeta functions of trinomial curves in terms of Gauss sums and Jacobi sums, and we obtain an explicit formula of the genus of a trinomial curve over a finite field, then we study the conditions for a trinomial curve to be a maximal curve over a finite field.

黎曼 zeta 函数无零点区域的最优余弦多项式

Optimal Cosine Polynomials for Riemann Zeta Zero-Free Region

本文用数值方法确定次数为 7 和 8 的最优余弦多项式，并给出黎曼 zeta 函数相应的无零点区域，推进了 Landau 构造下最大无零点区域问题的研究。

The Lerch Zeta Function II. Analytic Continuation

This is the second of four papers that study algebraic and analytic structures associated with the Lerch zeta function. In this paper we analytically continue it as a function of three complex variables. We that it is well defined as a multivalued function on the manifold M equal to C^3 with the hyperplanes corresponding to integer values of the two variables a and c removed. We show that it becomes single valued on the maximal abelian cover of M. We compute the monodromy functions describing the multivalued nature of this function on M, and determine various of their properties.

Converse theorems: from the Riemann zeta function to the Selberg class

This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions.

Hasse-Weil Zeta 函数模素数

Hasse-Weil Zeta Functions Modulo a Prime

本文通过有限域上分离有限型概形的有限态射，确定了在素数阶伽罗瓦群作用下zeta函数的模r约化，并应用于曲线及超椭圆/超椭圆曲线的分子。

Iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and a class of relations among multiple zeta values

In this paper we consider iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and define a class of $\mathbb{Q}$-linear relations among them, which arises from the differential structure of the iterated integrals with respect to $z$. We then define a new class of $\mathbb{Q}$-linear relations among the multiple zeta values by taking their limits of $z\rightarrow1$, which we call \emph{confluence relations} (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust …

Zeta functions of supersingular curves of genus 2

We determine what isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and to count the number of curves, up to k-isomorphism, leading to the same zeta function.

双曲平面上二次格的Zeta函数

Zeta functions of quadratic lattices of a hyperbolic plane

本文研究了双曲平面上二次格满秩子格真等价类的Dirichlet级数，推导出相关Zeta函数的显式公式与组合计算方法，并揭示了大量真类为单格类的解析性质。

Higher Selberg zeta functions for congruence subgroups

As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we determine the gamma factor explicitly in terms of the Barnes triple gamma function and the higher Dirichlet $L$-function.

On values of the higher derivatives of the Barnes zeta function at non-positive integers

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s ζ_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $ζ_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-p…

Cyclic relation for multiple zeta functions

The cyclic relation obtained in a study by Hirose, Murakami, and the first-named author, is a wide class of relations, which includes the well-known cyclic sum formula for multiple zeta and zeta-star values, and the derivation relation for multiple zeta values. In this paper, we present its generalization to complex variables. Our proof includes a new proof of the cyclic relation.

On the zeros of generalized Hurwitz zeta functions

In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function.

An Euler-type formula for $β(2n)$ and closed-form expressions for a class of zeta series

In a recent work, Dancs and He found an Euler-type formula for $\,ζ{(2\,n+1)}$, $\,n\,$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in comparison to $ζ(2n)$, which is a rational multiple of $π^{2n}$. For the Dirichlet beta function, the things are `inverse': $β(2n+1)$ is a rational multiple of $π^{2n+1}$ and no closed-form expression is known for $β(2n)$. Here in this work, I modify the Dancs-He approach in order to derive an Euler-type formula for $\,β{(2n)}$, including $\,β{(2)} = G$, the Cata…

Barnes multiple zeta-functions, Ramanujan's formula, and relevant series involving hyperbolic functions

In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan's classical formula for the Riemann zeta values can be derived from functional equations for Barnes zeta-functions. In the latter half part, we generalize some evaluation formulas of certain series involving hyperbolic functions in terms of Bernoulli polynomials. The original formulas were classically given by Cauchy, Mellin, Ramanujan, and later recovered and formulated by Berndt. From our consideration, we give multiple ve…