Kanetomo Sato
In this paper, we give an approach to the zeta values of a (proper regular) arithmetic scheme X at the integers r>=d:=dim(X), using étale cohomology of X with Q_p(r) and Z_p(r)-coefficients.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove an easy but interesting result about the linear independence of multiple zeta values of different weights.
We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.
We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for F_q[theta] and provide complete algebraic independence results for them.
Arithmetic sums and products of infinite multiple zeta-star values
本文研究了受限索引的无穷多重ζ星值的算术和与积，并受连分数理论和康托集启发，提出关于特定索引的ζ星值代数点及算术运算的猜想。
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.
We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a single prime p in time p^(1/2+o(1)), and another algorithm that computes zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.
In this work, we study the arithmetic nature of the numbers of the form $n^{\g}$, for $n \in \N$ and $\g\in \C$. We also consider a related conjecture and we show that it is a consequence of the unipresent Schanuel's conjecture.
In this paper, we partially extend recent results of Wan concerning the relationship between the zeta functions of a Calabi-Yau hypersurface and its (singular) mirror variety.
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.
We study the special value conjecture for the Zeta function of a proper regular arithmetic scheme X introduced by Flach and Morin in the case n=1. We compute the correction factor C(X,1) left unspecified in the original statement of the Flach-Morin Conjecture, thereby developing some results on the eh-topology introduced by Geisser. We then specialize further to the case where X is an arithmetic surface and show that the conjecture of Flach and Morin is equivalent to the Birch and Swinnerton-Dyer Conjecture.
Hyperderivatives of the deformation series associated with arithmetic gamma values and characteristic $p$ multiple zeta values
本文针对正特征数论中的特殊值问题，推广已有研究结果，借助多种代数工具证明了特定算术伽马值、受限正特征多重zeta值及其形变超导数的代数独立性。
A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gassmann-equivalence (a purely group theoretical property), but this statement fails if the degree exceeds the characteristic. We introduce a `T…
Mean-square values of the Riemann zeta function on arithmetic progressions
本文研究了黎曼zeta函数在等差数列上的二阶离散矩，推导出其渐近公式，揭示了离散矩与连续矩的关系，并指出当a为正整数时主项相等。
We investigate the properties of the zeta-function of well-rounded sublattices of a fixed arithmetic lattice in the plane. In particular, we show that this function has abscissa of convergence at $s=1$ with a real pole of order 2, improving upon a recent result of S. Kuehnlein. We use this result to show that the number of well-rounded sublattices of a planar arithmetic lattice of index less or equal $N$ is $O(N \log N)$ as $N \to \infty$. To obtain these results, we produce a description of integral well-rounded sublattices of a fixed planar integral well-rounded lattice and investigate conve…
We show that the recent conjecture of the first-named author for the special value at $s=1$ of the zeta function of an arithmetic surface is equivalent to the Birch-Swinnerton-Dyer conjecture for the Jacobian of the generic fibre.
In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL(2,Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem, the asymptotic behavior of the sum of the class number.
Universality theorems of the Selberg zeta functions for arithmetic groups
本文将 Selberg zeta 函数的普适性定理从模群及其主同余子群进一步推广到模群的一般子群和由不定四元数代数导出的余紧算术群，并得到了比已有结果更宽的成立区域。
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。
We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression $1/2 + i(an + b)$ with $a > 0$, $b$ real, exhibits a remarkable correspondance with the analogous continuous average and derive several consequences. For example, motivated by the linear independence conjecture, we show at least one third of the elements in the arithmetic progression $a n + b$ are not the ordinates of some zero of $ζ(s)$ lying on the critical line. This improves on earlier work of Martin and Ng. We then complement this result by producing large and small values of $ζ(s…
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $ζ(2)$ and $ζ(3)$, as well as to explain Rivoal's "infinitely-many" result (math.NT/0008051) and to prove that at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$, and $ζ(11)$ is irrational.
In this paper, we study the relation between the zeta function of a Calabi-Yau hypersurface and the zeta function of its mirror. Two types of arithmetic relations are discovered. This motivates us to formulate two general arithmetic mirror conjectures for the zeta functions of a mirror pair of Calabi-Yau manifolds.
In this paper, we define new fractal dimensions of a metric space in order to calculate the roughness of a set on a large scale. These fractal dimensions are called upper zeta dimension and lower zeta dimension. The upper zeta dimension is an extension of the zeta dimension introduced by Doty, Gu, Lutz, Elvira, Mayordomo, and Moser. We show that the upper zeta dimension is always a lower bound for the Assouad dimension. Moreover, we apply the upper zeta dimension to the existence of weak arithmetic patches of a given set. Arithmetic patches are higher dimensionalized arithmetic progressions. A…
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.
We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.
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。
In this paper, we study the arithmetic zeta function $$\mathscr{Z}_{\mathcal{X}}(s) = \prod_p \prod_{\substack{x \in \mathcal{X}_p \\ \text{closed}}} \Big( \frac{1}{1-|κ(x)|^{-s}} \Big)^{\mathfrak{m}_{p}(x)}$$ associated to a scheme $\mathcal{X}$ of finite type over $\mathbb{Z}$, where $κ(x)$ denotes the residue field and $\mathfrak{m}_{p}(x)$ the multiplicity of $x$ in $\mathcal{X}_p$. If $\mathcal{X}$ is defined over a finite field, then $\mathscr{Z}_{\mathcal{X}}$ appears naturally in the context of point counting with multiplicities. We prove that $\mathscr{Z}_{\mathcal{X}}$ admits a merom…
We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms of Weil-étale cohomology complexes. This extends work of Lichtenbaum \cite{Lichtenbaum05} and Geisser \cite{Geisser04b} for $\mathcal{X}$ of characteristic $p$, of Lichtenbaum \cite{li04} for $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$ and $n=0$ where $F$ is a number field, and of the second author for arbitrary $\mathcal{X}$ and $n=0$ \cite{Morin14}. We show that our conjecture is compatible with the Tamaga…
In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some $\mathbb{Q}$-rational differential forms on the Fermat curve $F_2$ of degree 2 and discuss their arithmetic properties. We also investigate a motivic structure of the motivic periods corresponding to our periods. However, in order to study them, the current theory for motivic zeta elements is insufficient, and it leads us to study the base extension of the space of the motivic periods $\mathcal{H}_4$ of level 4 and its Galois invariant part.
Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we obtain (unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing …
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.
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.
The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading factors of the infinite product over zeta-functions. If rooted at the Dirichlet series for powers, for sums-of-divisors and for Euler's totient, the inheritance of multiplicativity through Dirichlet convolution or ordinary multiplication of pairs of arithmetic functions generates most of the results.
We prove an approximation to an Ihara-formula for the zeta function of an arithmetic quotient of the Bruhat-Tits building of a p-adic PGL(3).
We use partial zeta functions to analyse the asymptotic behaviour of certain smooth arithmetical sums over smooth k-free integers.
A Riemmanian foliated dynamical system of 3-dimension $(\mathrm{RFDS}^{3})$ is a closed Riemannian 3-manifold with additional structures: foliation, dynamical system. In the context of arithmetic topology, it is a geometric/analytic analogue of an arithmetic scheme with a conjectural dynamical system suggested by C. Deninger. In this paper, we show leafwise cohomological expression of dynamical zeta function on a Riemannian foliated dynamical system.
Zeta-values of one-dimensional arithmetic schemes at strictly negative integers
本文通过Weil-étale形式化，推导了一维算术方案在严格负整数处的ζ值公式，并将其与主要猜想联系起来。
We point out analogies between (a) the explicit formulas in analytic number theory and transversal index theory, (b) Lichtenbaum's recent conjectures on special values of Hasse-Weil zeta functions and a formula for special values of Ruelle zeta functions, (c) work of Cramer on the zeroes of the Riemann zeta function and a result of Chazarain on the trace of a wave operator. These analogies suggest certain problems in the analysis on foliated spaces. The paper is mostly a review of previous work but the ideas concerning (c) are new.
Weil-étale cohomology and zeta-values of arithmetic schemes at negative integers
本文基于韦尔-étale上同调提出关于算术方案在负整数处ζ函数特例值的猜想，并证明其在特定方案上成立。
In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the ``explicit formula'' of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence…
Answering problems of Manin, we use the critical $L$-values of even weight $k\geq 4$ newforms $f\in S_k(Γ_0(N))$ to define zeta-polynomials $Z_f(s)$ which satisfy the functional equation $Z_f(s)=\pm Z_f(1-s)$, and which obey the Riemann Hypothesis: if $Z_f(ρ)=0$, then $\operatorname{Re}(ρ)=1/2$. The zeros of the $Z_f(s)$ on the critical line in $t$-aspect are distributed in a manner which is somewhat analogous to those of classical zeta-functions. These polynomials are assembled using (signed) Stirling numbers and "weighted moments" of critical values $L$-values. In analogy with Ehrhart polyno…
We consider zeta functions: $Z(f ;P ;s)=\sum_{\m \in \N^{n}} f(m_1,..., m_n) P(m_1,..., m_n)^{-s/d}$ where $P \in \R [X_1,..., X_n]$ has degree $d$ and $f$ is a function arithmetic in origin, e.g. a multiplicative function. In this paper, I study the meromorphic continuation of such series beyond an a priori domain of absolute convergence when $f$ and $P$ satisfy properties one typically meets in applications. As a result, I prove an explicit asymptotic for a general class of lattice point problems subject to arithmetic constraints.
Spencer Bloch and the author formulated a general conjecture (Tamagawa number conjecture) on the relation between values of zeta functions of motives and arithmetic groups associated to motives. We discuss this conjecture, and describe some application of the philosophy of the conjecture to the study of elliptic curves.
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…
We prove that the special value conjecture for the Zeta function of a proper, regular arithmetic scheme X that we formulated in our previous article [8] is compatible with the functional equation of the Zeta function provided that the factor C(X,n) we were not able to compute in loc. cit. has the simple explicit form suggested in [9].
We study the asymptotics of Hankel determinants constructed using the values $ζ(an+b)$ of the Riemann zeta function at positive integers in an arithmetic progression. Our principal result is a Diophantine application of the asymptotics.
This paper is dedicated to the description of the poles of the Igusa local zeta functions $Z(s,f,v)$ when $f(x,y)$ satisfies a new non degeneracy condition, that we have called arithmetic non degeneracy. More precisely, we attach to each polynomial $f(x,y)$, a collection of convex sets $Γ^{A}$ called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non degeneracy with respect to $Γ^{A}(f)$. The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials …
In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.
We obtain the last of the standard Kuznetsov formulas for $SL(3,\Bbb{Z})$. In the previous paper, we were able to exploit the relationship between the positive-sign Bessel function and the Whittaker function to apply Wallach's Whittaker expansion; now we demonstrate the expansion of functions into Bessel functions for all four signs, generalizing Wallach's theorem for $SL(3)$. As applications, we again consider the Kloosterman zeta functions and smooth sums of Kloosterman sums. The new Kloosterman zeta functions pose the same difficulties as we saw with the positive-sign case, but for the nega…
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series -- as well as "Eulerian" $q$-hypergeometric series -- enjoying many interesting relations. We find a number of theorems of classical numb…
The volume of the unit sphere in every dimension is given a new interpretation as a product of special values of the zeta function of $\mathbb{Z}$, akin to volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A product formula is found for this zeta function that specializes to Catalan numbers. Moreover, certain closed-form expressions for various other zeta values are deduced, in particular leading to an alternative perspective on Euler's values of the Riemann zeta function.
The article is dedicated to the memory of George Voronoi. It is concerned with ($p$-adic) $L$-functions (in partially ($p$-adic) zeta functions) and cyclotomic ($p$-adic) (multiple) zeta values. The beginning of the article contains a short summary of the results on the Bernoulli numbers associated with the studies of George Voronoi. Results on multiple zeta values have presented by D. Zagier, by P. Deligne and A.Goncharov, by A. Goncharov, by F. Brown, by C. Glanois and others. S. Ünver have investigated p-adic multiple zeta values in the depth two. Tannakian interpretation of p-adic multiple…
It is pointed out that the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page $332$ of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters $N$ and $h$ that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for $N$ odd and $m>0$, gives a relation between $ζ(2m+1)$ a…
Vertex volumes, lattice-minima tails, and height zeta functions for the standard arithmetic quotient of $\operatorname{PGL}_d$
该研究针对 $\text{PGL}_d$ 的标准算术商，基于建筑理论计算顶点体积，引入同伦不变归一性格极小值高度，确定其可积阈值与尖点尾估计，推导高度泽塔函数的收敛域与解析延拓性质，明确计算了 $d=3,4,5$ 时的有理函数。
We study mirror symmetric pairs of Calabi--Yau manifolds over finite fields. In particular we compute the number of rational points of the manifolds as a function of the complex structure parameters. The data of the number of rational points of a Calabi--Yau $X/\mathbb{F}_q$ can be encoded in a generating function known as the congruent zeta function. The Weil Conjectures (proved in the 1970s) show that for smooth varieties, these functions take a very interesting form in terms of the Betti numbers of the variety. This has interesting implications for mirror symmetry, as mirror symmetry exchan…
Multiple zeta values with varying constant fields
本文利用 Papanikolas 的 $t$-动机理论，证明了具有不同常数域的函数域上的多重 zeta 值之间不存在代数关系。
On arithmetic nature of $q$-analogue of the generalized Stieltjes constants
本文拓展了2003年关于Hurwitz zeta函数$q$-模拟的研究，推导了其洛朗级数前两项系数的闭式，探究了相关超越性，提出了线性无关猜想并证明了超半数相关数在有理数域上线性无关。
The partial fraction expansion of coth($π$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet series is also proposed. The resulting formulas are new, as far as we know, since they could not be found in any of the classical or recent handbooks of formulas that were at our disposal.
This paper is concerned with the arithmetic of the elliptic K3 surface with configuration [1,1,1,12,3*]. We determine the newforms and zeta-functions associated to X and its twists. We verify conjectures of Tate and Shioda for the reductions of X at 2 and 3.
It is a classic result that two number fields have equal Dedekind zeta functions if and only if the arithmetic type of a prime $p$ is the same in both fields for almost all prime $p$. Here, almost all means with the possible exception of a set of Dirichlet density zero. One of the results of this paper shows that the condition density zero can be improved to a specific positive density that depends solely in the degree of the fields. More specifically, for every positive $n$ we exhibit a positive constan $c_{n}$ such that any two degree $n$ number fields $K$ and $L$ are arithmetically equivale…
Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function
本文仿照Ishibashi方法推导Zagier型zeta函数$\tilde{\mathcal{Z}}$在$s=1$处的全部Laurent系数，并研究其中涉及的函数$\mathfrak{F}_k^0(x)$的函数方程、Hecke算子作用及积分联系。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。