Zeta Correspondences in Rank-n

On the Rankin-Selberg zeta-function

We obtain the approximate functional equation for the Rankin-Selberg zeta-function on the 1/2-line.

On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case…

Geometric zeta functions for higher rank p-adic groups

The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions

Elliptic curves of high rank and the Riemann zeta function on the one line

We describe some experiments that show a connection between elliptic curves of high rank and the Riemann zeta function on the one line. We also discuss a couple of statistics involving $L$-functions where the zeta function on the one line plays a prominent role.

多重ζ星值集合的拓扑

The topology of the set of multiple zeta-star values

本文通过多重积分表示研究多重ζ星值集合的拓扑结构，揭示其序关系和稠密性，并计算了相关维度。

亏格2曲线的非阿贝尔zeta函数的黎曼猜想

Riemann Hypothesis for Non-Abelian Zeta Functions of Genus 2 Curves

本文针对有限域上亏格2曲线的Weng非阿贝尔zeta函数，利用半稳定丛模空间几何性质、高秩结果与渐近分析，证明了当秩n趋于无穷时黎曼猜想成立。

Mahler/Zeta Correspondence

The Mahler measure was introduced by Mahler in the study of number theory. It is known that the Mahler measure appears in different areas of mathematics and physics. On the other hand, we have been investigated a new class of zeta functions for various kinds of walks including quantum walks by a series of our previous work on "Zeta Correspondence". The quantum walk is a quantum counterpart of the random walk. In this paper, we present a new relation between the Mahler measure and our zeta function for quantum walks. Firstly we consider this relation in the case of one-dimensional quantum walks…

The Riemann hypothesis for Weng's zeta function of $Sp(4)$ over $\mathbb{Q}$

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple poles. The rank one zeta function is the Dedekind zeta function. For the rank two case, the Riemann hypothesis is proved for a general number field. Recently, he defined more general new zeta function associated to a pair of reductive group and its maximal parabolic subgroup. As well as high rank zeta functions, the new zeta function satisfies standard prope…

Zeta morphisms for rank two universal deformations

In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato's zeta morphisms for Galois representations associated to Hecke eigen cusp newforms. The existence of such morphisms was predicted by Kato's generalized Iwasawa main conjecture. Based on Kato's original construction, we construct our zeta morphisms using many deep results in the theory of p-adic (local and global) Langlands correspondence for GL_{2/Q}. As an app…

奇异曲线的Motivic Coh与Quot zeta函数

Motivic Coh and Quot zeta functions of singular curves

本文建立代数框架研究奇异曲线上Quot与Coh zeta函数，证明其motivic有理性和反射原理，并揭示与Rogers--Ramanujan型q-级数的新联系。

Two-variable zeta functions and regularized products

In this paper we prove a regularized product expansion for the two-variable zeta functions of number fields introduced by van der Geer and Schoof. The proof is based on a general criterion for zeta-regularizability due to Illies. For number fields of non-zero unit rank our method involves a result of independent interest about the asymptotic behaviour of certain oscillatory integrals in the geometry of numbers. We also explain the cohomological motivation for the paper.

Mean-periodicity and zeta functions

This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of arithmetic scheme with its expected analytic shape is shown to correspond to mean-periodicity of a certain explicitly defined function associated to the zeta function. This correspond…

Zeta Functions on Arithmetic Surfaces

We use a form of lifted harmonic analysis to develop a two-dimensional adelic integral representation of the zeta functions of simple arithmetic surfaces. Manipulations of this integral then lead to an adelic interpretation of the so-called mean-periodicity correspondence, which is comparable to the better known automorphicity conjectures for the generic fibre.

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。

Functional equations for zeta functions of $\mathbb{F}_1$-schemes

For a scheme $X$ whose $\mathbb F_q$-rational points are counted by a polynomial $N(q)=\sum a_iq^i$, the $\mathbb{F}_1$-zeta function is defined as $ζ(s)=\prod(s-i)^{-a_i}$. Define $χ=N(1)$. In this paper we show that if $X$ is a smooth projective scheme, then its $\mathbb{F}_1$-zeta function satisfies the functional equation $ζ(n-s) = (-1)^χζ(s)$. We further show that the $\mathbb{F}_1$-zeta function $ζ(s)$ of a split reductive group scheme $G$ of rank $r$ with $N$ positive roots satisfies the functional equation $ζ(r+N-s) = (-1)^χ( ζ(s) )^{(-1)^r}$.

zeta-star 对应的导数函数

The derivative function of the zeta-star correspondence

本文研究 zeta-star 对应的右导数，证明其连续性、跳跃与维数性质，并给出 Hölder 指数与 Hausdorff 谱的线性结果。

The zeta-function of monomial deformations of Fermat hypersurfaces

This paper intends to give a mathematical explanation for results on the zeta-function of some families of varieties recently obtained in the context of Mirror Symmetry. In doing so, we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in P^n. In particular, we extend the monomial-motive correspondence of Kadir and Yui and we give explicit solutions to the p-adic Picard-Fuchs equation associated with monomial deformations of Fermat hypersurfaces. As a by-product we obtain Poincare Duality for the Rigid cohomology of certa…

Rank one case of Dwork's conjecture

This paper proves the general rank one case of Dwork's conjecture over the affine space. It generalizes and improves the method of ANT-0141 "Dwork's conjecture on unit root zeta functions" (Ann. Math., 150(1999), 867-929). In addition, explicit information about the zeros and poles (along the Gouvêa-Mazur conjecture direction) for the unit root zeta function is obtained. The paper is to appear in JAMS.

完全奇深度分级多重zeta值与周期多项式

Totally odd depth-graded multiple zeta values and period polynomials

本文研究完全奇动机深度分级多重zeta值的关系，确定Brown矩阵的秩，给出$C_{N,3}$和$C_{N,4}$秩上界的新证明，并提出递归方法将一般问题归结为同构猜想。

On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces

We derive approximate formulas for the logarithmic de- rivative of the Selberg and Ruelle zeta functions over compact, even- dimensional, locally symmetric spaces of rank one. The obtained for- mulas are given in terms of the zeta-singularities.

Simple functional equations for generalized Selberg zeta functions with Tate motives

We prove that for a compact locally symmetric Riemannian space $M$ of rank 1 there exist infinitely many automorphic Tate motives $f$ such that the generalized Selberg zeta function $Z_{M(f)}(s)$ satisfies a simple functional equation in the sense that it has no gamma factors.

Order of zeta functions for compact even-dimensional symmetric spaces

Some zeta functions which are naturally attached to the locally homogeneous vector bundles over compact locally symmetric spaces of rank one are investigated. We prove that such functions can be expressed in terms of entire functions whose order is not larger than the dimension of the corresponding compact, even-dimensional, locally symmetric space.

具有无理值特征标的表代数与融合环的zeta函数

Zeta functions for table algebras and fusion rings with irrational-valued characters

本文针对具有无理值不可约特征标的秩3标准整表代数与整融合环，通过显式计算zeta积分，求出了相应序的zeta函数。

Statistics for ordinary Artin-Schreier covers and other $p$-rank strata

We study the distribution of the number of points and of the zeroes of the zeta function in different $p$-rank strata of Artin-Schreier covers over $\F_q$ when $q$ is fixed and the genus goes to infinity. The $p$-rank strata considered include the ordinary family, the whole family, and the family of curves with $p$-rank equal to $p-1.$ While the zeta zeroes always approach the standard Gaussian distribution, the number of points over $\F_q$ has a distribution that varies with the specific family.

Rationality of the zeta function of the subgroups of abelian $p$-groups

Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $σ_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the corresponding zeta-function on groups of $p$-rank bounded by some constant $r$ is rational with a simple denominator. We also provide two explicit examples in rank $r=3$ and $r=4$ as well as a closed formula for $σ_a(F)$.

关于正特征中任意秩多重爱森斯坦级数的q-洗牌关系

On $q$-Shuffle Relations for Multiple Eisenstein Series of Arbitrary Rank in Positive Characteristic

本文在正特征中定义任意秩多重爱森斯坦级数，并证明其满足与多重zeta值相同的q-洗牌关系，将关系从值提升到函数。

GL₂×GL₂秩纳积分的积分性：对于分裂表示

Integrality of $\mathrm{GL}_2\times\mathrm{GL}_2$ Rankin-Selberg integrals for ramified representations

本文研究GL₂×GL₂秩纳积分在分裂表示下的积分性，引入了(π₁×π₂)积分数据的概念，并改进了雅克布-兰格兰德的GCD结果，与新形式的傅里叶系数的积分性相容。

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.

数域上费马曲面的伽罗瓦不变奈龙 - 塞维里秩：伽罗瓦模、封闭形式、阈值及精确表格

Galois-invariant Néron--Severi ranks of Fermat surfaces over number fields: a Galois module, closed forms, a threshold, and exact tables

研究数域上费马曲面的伽罗瓦不变奈龙 - 塞维里秩，通过显式特征平均计算，给出\(\gcd(d,6)=1\)时封闭形式及\(4\leq d\leq30\)的有理秩，证明相关公式，建立阈值和轨道规则，给出精确表格及平均阶，结果无条件且有辅助文件支持。

Non-Abelian L Functions for Function Fields

This is an integrated part of our Geo-Arithmetic Program. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields by a weighted count of semi-stable bundles. Basic properties such as rationality and functional equation are established. Examples of rank two zetas over genus two curves are given as well. Based on this and motivated by our study for non-abelian zetas of number fields, general non-abelian $L$ functions for function fields are defined and studied using Langlands and Morris' theory of Eisenstein series.

利用局部ζ积分计算表代数序的ζ函数

Computing zeta functions of table algebra orders using local zeta integrals

针对Solomon初等方法计算秩≥3的整表代数序的ζ函数存在困难的问题，研究了用Bushnell和Reiner的局部ζ积分法计算这类序的显式ζ函数，并给出了应用示例。

Bi-algebraicity in the rank one Riemann--Hilbert correspondence via o-minimality

For a smooth, projective, complex algebraic variety $X$, the Riemann--Hilbert correspondence establishes a complex analytic isomorphism between the `Betti moduli space' of rank $n$ local systems on $X^\mathrm{an}$ and the `de Rham moduli space' of rank $n$ vector bundles with flat connection on $X$. In the rank one case, C. Simpson precisely characterizes the subvarieties of these moduli spaces that are `bi-algebraic' for this typically transcendental, analytic isomorphism. In this short note, we give a new proof of this characterization of Simpson, using methods from o-minimal geometry. We ad…

Ramanujan series upside-down

We prove that there is a correspondence between Ramanujan-type formulas for 1/π, and formulas for Dirichlet L-values. The same method also allows us to resolve certain values of the Epstein zeta function in terms of rapidly converging hypergeometric functions. The Epstein zeta functions were previously studied by Glasser and Zucker.

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。

Elliptic curves with large rank over function fields

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic curves over number fields.

Iwasawa theory and zeta elements for $\mathbb{G}_m$

We describe an explicit `higher rank' Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of general number fields. We then show that this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture. As a first application of this approach, we use it to prove new cases of the conjecture for Tate motives over natural families of abelian CM extensions of totally real fields for which the relevant $p$-adic $L$-functions possess trivial zeroes.

Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces

We present the first example of the Selberg type zeta function for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real quadratic field. We show that they have meromorphic extensions to the whole complex plane and satisfy functional equations. The method is based on considering the differences among several Selberg trace formulas with different weights for the Hilbert modular group. Besides as an application of the differences of the Selberg trace formula, we also obtain an a…

Test vectors for Rankin-Selberg $L$-functions

We study the local zeta integrals attached to a pair of generic representations $(π,τ)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $π$ and $τ$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.

量子细胞自动机zeta函数的绝对zeta函数

Absolute zeta functions for zeta functions of quantum cellular automata

本文证明了量子细胞自动机（QCA）给出的新zeta函数是权取决于构型空间大小的绝对自守形式，并以张量型QCA为例，用多重gamma函数表示其绝对zeta函数，进而通过多重sine函数得到其函数方程。

Non-Abelian Zeta Function, Fokker-Planck Equation and Projectively Flat Connection

Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over the moduli spaces, all these are connected with the zeros of rank $n$ non-abelian zeta functions of the field of rationals, which are known lie on the central line except a finitely many if $n\geq 2$. Certainly, when $n=1$, our current work recovers that of Armitage, which from th…

Barnes多重zeta函数及相关问题的Lindelöf假设类比

Analogues of the Lindelöf Hypothesis for the Barnes multiple zeta function and related problems

研究Barnes多重zeta函数的Lindelöf假设类比，通过积分均值建立等价条件，并揭示依赖于有理秩的独特现象。

SL_2 的元循环覆盖的局部 zeta 函数的函数方程与 gamma 因子

Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL_2

本文为 SL_2 的元循环二重覆盖上的超尖点表示引入局部 zeta 函数，证明其函数方程并用 Bessel 函数的 Mellin 变换给出 gamma 因子，进而通过局部 theta 对应将其与 PGL_2 上二元对称矩阵预齐性向量空间的球面函数 zeta 积分等同起来。

Special zeta values using tensor powers of Drinfeld modules

We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of vector valued Anderson generating functions, we give formulas for the coefficients of the logarithm and exponential functions associated to these A-modules. We then show that there exists a vector whose bottom coordinate contains a Goss zeta value, whose evaluation under the exponential function is defined over the Hilbert class field. This allows us to prove the transcendence of Goss zeta values and periods of Drinfeld modules as wel…

Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

We study generalised prime systems $\mathcal{P}$ $(1<p_1\leq p_2\leq...,$ with $p_j\in\R$ tending to infinity) and the associated Beurling zeta function $ζ_{\mathcal{P}}(s) =\prod_{j=1}^{\infty} (1-p_j^{-s})^{-1}$. Under appropriate assumptions, we establish various analytic properties of $ζ_{\mathcal{P}}(s)$, including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of $ζ_{\math…

Universal二次型与Dedekind zeta函数

Universal quadratic forms and Dedekind zeta functions

本文利用Dedekind zeta函数研究全实数域上的universal二次型，在余微商由全正元素生成的假设下给出秩的显式上界，并探讨去除该假设的路径。

Results on zeta functions for codes

We give a new and short proof of the Mallows-Sloane upper bound for self-dual codes. We formulate a version of Greene's theorem for normalized weight enumerators. We relate normalized rank-generating polynomials to two-variable zeta functions. And we show that a self-dual code has the Clifford property, but that the same property does not hold in general for formally self-dual codes.

The cotype zeta function of $\mathbb{Z}^d$

We give an asymptotic formula for the number of sublattices $Λ\subseteq \mathbb{Z}^d$ of index at most $X$ for which $\mathbb{Z}^d/Λ$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to recent work of Stanley and Wang on Smith Normal Forms of random integral matrices and discuss connections to the Cohen-Lenstra heuristics. Our arguments are based on Petrogradsky's formulas for the cotype zeta function of $\mathbb{Z}^d$, a multivariable generalization of the subgroup growth zeta function of $\mathbb{Z}^d$.

Möbius functions of higher rank and Dirichlet series

We introduce Möbius functions of higher rank, a new class of arithmetic functions so that the classical Möbius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role.

关于半稳定椭圆曲线的伴随Selmer群与Flach的zeta元素

On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements

本文显式构造了半稳定椭圆曲线的秩1本原Stark系统，证明了其与极小模性提升定理的等价性，并比传统几何Euler系统方法更精细地解释了Flach的zeta元素。

On mean values of some zeta-functions in the critical strip

For a fixed integer $k\ge 3$ and fixed $1/2 < σ> 1$ we consider $$ \int_1^T |ζ(σ+ it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2σ}T + R(k,σ;T), $$ where $R(k,σ;T) = o(T) (T\to\infty)$ is the error term in the above asymptotic formula. Hitherto the sharpest bounds for $R(k,σ;T)$ are given for certain ranges of $σ$. We also obtain new mean value results for the zeta-functions of holomorphic cusp forms and the Rankin-Selberg series.

Ruelle type L-functions versus determinants of Laplacians for torsion free abelian groups

We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus $\bR^{\n}/\G$ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.

On Zeta Functions and Families of Siegel Modular Forms

Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modu…

Existence of a zero-free strip for the Riemann zeta function

In a recent article a breakthrough was made in quantum computer science which established the existence of a phenomenon commonly known as $\mathsf{MIP}^\ast = \mathsf{RE}$. We show that the existence of this phenomenon implies that the supremum of the real parts of the zeroes of the Riemann zeta function is less than one. Our main tool is an assertion about the representation theory of the rank two free group that we refer to as the charged mean ergodic theorem. Our method also involves the construction of novel modular functions we refer to as complete electromagnetic functions.

A One Parameter Family of Calabi-Yau Manifolds with Attractor Points of Rank Two

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over Q this mo…