On the local factor of the zeta function of quadratic orders

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。

Shuffle Product for Multiple Dedekind Zeta Values over Imaginary Quadratic Fields

Multiple Dedekind zeta values were recently defined by the second author. In a separate paper, the second author constructed double shuffle relations in some cases as a response to questions asked by Richard Hain and Alexander Goncharov. In this paper, we develop the technique for obtaining more shuffle relations and produce many examples of shuffle products over an imaginary quadratic field. We also define the notion of self shuffle of a (multiple) Dedekind zeta value and use it at many instances. We define a refinement of the multiple Dedekind zeta values. Our key examples are self shuffles …

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。

某些二次序的Zeta函数

Zeta Functions of Certain Quadratic Orders

本文通过递归方法计算了特定二次序族zeta函数的显式公式，为后续论文证明Arthur关于轨道积分乘积公式的预测奠定了基础。

Polynomial behavior of special values of partial zeta function of real quadratic fields at s=0

We compute the special values of partial zeta function at $s=0$ for family of real quadratic fields $K_n$ and ray class ideals $\fb_n$ such that $\fb_n^{-1} = [1,δ(n)]$ where the continued fraction expansion of $δ(n)$ is purely periodic and each terms are polynomial in $n$ of bounded degree $d$. With an additional assumptions, we prove that the special values of partial zeta function at $s=0$ behaves as quasi-polynomial. We apply this to obtain that the special values the Hecke's $L$-functions at $s=0$ for a family of for a Dirichlet character $χ$ behave as quasi-polynomial as well. We compute…

枚举二次形式子形式的 Zeta 函数

Zeta functions enumerating subforms of quadratic forms

本文构造枚举二次形式满秩子形式等价类的 Dirichlet 级数，建立其与相关虚二次域理想类计数级数的联系，并给出方格、六角格等情形的显式公式及解析性质。

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…

Special values of partial zeta functions of real quadratic fields at nonpositive integers and Euler-Maclaurin formula

We compute the special values at nonpositive integers of the partial zeta function of an ideal of a real quadratic field applying an asymptotic version of Euler-Maclaurin formula to the lattice cone associated to the ideal considered. The Euler-Maclaurin formula involved is obtained by applying the Todd series of differential operators to an integral of a small perturbation of thecone. The additive property of Todd series w.r.t. the cone decomposition enables us to express the partial zeta values in terms of the continued fraction of the reduced element of the ideal. The expression obtained us…

Explicit identities on zeta values over imaginary quadratic field

In this article, we study special values of the Dedekind zeta function over an imaginary quadratic field. The values of the Dedekind zeta function at any even integer over any totally real number field is quite well known in literature. In fact, in one of the famous article, Zagier obtained an explicit formula for Dedekind zeta function at point 2 and conjectured an identity at any even values over any number field. We here exhibit the identities for both even and odd values of the Dedekind zeta function over an imaginary quadratic field which are analogous to Ramanujan's identities for even a…

A New Generating Function for Calculating the Igusa Local Zeta Function

A new method is devised for calculating the Igusa local zeta function $Z_f$ of a polynomial $f(x_1,\dots,x_n)$ over a $p$-adic field. This involves a new kind of generating function $G_f$ that is the projective limit of a family of generating functions, and contains more data than $Z_f$. This $G_f$ resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitl…

A quadratic divisor problem and moments of the Riemann zeta-function

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta…

Hyperbolic period of the nonholomorphic Eisenstein series

This note contains an attempt to relate Hecke's presentation of an ideal class zeta function in a real quadratic field as an integral of the nonholomorphic Eisenstein series along the loop on modular curve and Zagier's decomposition of this zeta function into the sum of zeta functions of indefinite quadratic forms. (See D.Zagier, A Kronecker limit formula for real quadratic field.) It is a preliminary version.

On the zeros of Epstein zeta functions

We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.

Zero-density estimates for Epstein zeta functions

We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients in the vertical strip $ σ_1 < \Re s < σ_2 $, where $ 1/2 < σ_1 < σ_2 < 1 $. When the class number of the quadratic form is bigger than 1, Voronin gives a lower bound and Lee gives an asymptotic formula for the number of zeros. In this paper, we improve their results by providing a new upper bound for the error term.

向量值周期多项式与二次域的ζ值

Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields

本文研究了二次域的ζ值与向量值周期多项式的关系，通过二元二次型构造了两种cusps形式，并利用闭合公式和对称性，将ζ值差异表示为伯努利数和二次型求和。

Quadratic Irrationals, Closed Geodesics on the Modular Surface and Dynamical Zeta Functions

We show that generating functions associated to the sequence of convergents of a quadratic irrational are related in a natural way to the dynam- ical zeta function of a hyperbolic automorphism of the 2-torus. As a corollary, this shows that the Lévy constant of a quadratic irrational appears naturally as the topological entropy of such maps.

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.

Zeta Functions for the Adjoint Action of GL(n) and density of residues of Dedekind zeta functions

We define zeta functions for the adjoint action of GL(n) on its Lie algebra and study their analytic properties. For n<4 we are able to fully analyse these functions, and recover the Shintani zeta function for the prehomogeneous vector space of binary quadratic forms for n=2. Our construction naturally yields a regularisation, which is necessary for the improvement of the properties of these zeta function, in particular for the analytic continuation if n>2. We further obtain upper and lower bounds on the mean value X^{-5/2}\sum_{E} \res_{s=1}ζ_E(s) as X->\infty, where E runs over totally real …

Hecke's integral formula for quadratic extensions of a number field

Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F, we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the 0-th Laurent coefficients at s=1 of zeta functions of K and F.

惰性二次序的Coh zeta函数

Coh zeta functions for inert quadratic orders

本研究针对一类惰性二次序的Coh zeta函数提出其可由$t$形变Bressoud $q$级数表示的猜想，通过推导相关显式公式提供有力证据，完善了二次序zeta函数与经典恒等式关联的研究，核心工具为偏序集Möbius反演新方法。

Moments of the Dedekind zeta function and other non-primitive L-functions

We give a conjecture for the moments of the Dedekind zeta function of a Galois extension via the hybrid product method. The moments of the product of primes are evaluated using the Montgomery-Vaughan mean value theorem whilst for the moments of the product over zeros we give a heuristic argument involving random matrix theory. The asymptotic for the first moment of the product over zeros is then proved for quadratic extensions. We are also able to reproduce our main conjecture in the quadratic case by using a modified version of the moments recipe. Finally, we generalise our methods to give a …

爱泼斯坦ζ函数在临界线上包含正比例的非平凡零点

The Epstein zeta-function contains a positive proportion of non-trivial zeros on the critical line

本文证明了对应整数系数二元正定二次型的爱泼斯坦ζ函数，其非平凡零点在临界线上占正比例，解决了该函数零点分布的关键问题。

虚二次数域中的显式公式

Explicit formula in a imaginary quadratic number field

本文将Haran基于Riesz位势、平等处理所有位的Riemann zeta函数显式公式推广到Q的虚二次扩张，通过定义完全分歧与复情形的Riesz核，揭示其与Dedekind zeta函数各位贡献的自然联系。

Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.

关于余切 zeta 有限 Dirichlet 级数的某一恒等式及其在 Berndt–Arakawa 公式中的应用

On a certain identity for the cotangent zeta finite Dirichlet series and its application to the Berndt--Arakawa formula

本文通过建立两个有限余切 Dirichlet 级数的渐近恒等式，为 Berndt–Arakawa 公式提供了一种新颖简洁的证明。

虚二次域的类数ζ函数

Class number zeta function of imaginary quadratic fields

本文引入一个通过类数计数虚二次数域的ζ函数，证明该函数仅依赖于八次单位根，从而得到素数类数的下界。方法基于动力系统周期点的研究。

Isotypic Decomposition of the Cohomology and Factorization of the Zeta Functions of Dwork Hypersurfaces

The aim of this article is to illustrate, on the example of Dwork hypersurfaces, how the study of the representation of a finite group of automorphisms of a hypersurface in its etale cohomology allows to factor its zeta function.

Epstein zeta-functions, subconvexity, and the purity conjecture

Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.

Rogers-Ramanujan连分数的对数矩与奇数zeta值

Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values

该研究探讨Rogers-Ramanujan连分数的尖点展开对其普通与对数Mellin矩的控制作用，证明相关系数的代数性质，确定归一化矩生成函数的极点，并建立反射恒等式等结果。

An analogue of Wilton's formula and values of Dedekind zeta functions

J. R. Wilton obtained an expression for the product of two Riemann zeta functions. This expression played a crucial role to find the approximate functional equation for the product of two Riemann zeta functions in the critical region. We find analogous expressions for the product of two Dedekind zeta functions and then use these expressions to find some expressions for Dedekind zeta values attached to arbitrary real as well as quadratic number fields at any positive integer.

On Kronecker limit formulas for real quadratic fields

Let $ζ(s,C)$ be the partial zeta function attached to a ray class C of a real quadratic field. We study this zeta function at s=1 and s=0, combining some ideas and methods due to Zagier and Shintani. The main results are (1) a generalization of Zagier's formula for the constant term of the Laurent expansion at s=1, (2) some expressions for the value and the first derivative at s=0, related to the theory of continued fractions, and (3) a simple description of the behavior of Shintani's invariant X(C), which is related to $ζ'(0,C)$, when we change the signature of C.

On the zeros of the Epstein zeta function

In this article, we count the number of consecutive zeros of the Epstein zeta-function, associated to a certain quadratic form, on the critical line with ordinates lying in $[0,T], T$ sufficiently large and which are separated apart by a given positive number $V$.

虚二次域上 $p$-ordinary 模形式的 Zeta 元素及其应用

Zeta elements for $p$-ordinary modular forms over imaginary quadratic fields and applications

本文在虚二次域上构造高权模形式的 zeta 元素并证明显式互反律，从而在弱假设下证明了 Kato 岩泽主猜想，推广了椭圆曲线情形。

Distribution of primes represented by polynomials and Multiple Dedekind zeta functions

n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for any irreducible polynomial $f$. Here, we formulate a conjecture for all irreducible polynomials. We also consider conjectures for distribution of pairs of primes. It show unexpected relation to multiple Dedekind zeta function - at $s=2$ for one prime and at $(s_1,s_2)=(2,2)$ for pairs of primes. We tested the conjecture for pairs of primes for several quadrat…

Schur函数第九变体的二次关系及其在Schur多重zeta函数中的应用

Quadratic relations for ninth variations of Schur functions and application to Schur multiple zeta functions

本文为Schur函数第九变体建立二次代数关系，并应用于研究对角常数Schur多重zeta函数在矩形形状下的特殊值。

二次覆盖的泽塔函数范畴化

Categorifying Zeta Functions for Quadratic Covers

针对数域二次扩张、代数曲线分歧双覆盖等四类二次覆盖情形，通过抽象关联代数的目标线性代数统一方法范畴化泽塔函数的L-函数乘积公式，并给出超椭圆曲线点计数、图素数分布、同源图圈计数及二次互反律分析等应用。

Indefinite zeta functions

We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray cla…

A modular relation involving non-trivial zeros of the Dedekind zeta function, and the Generalized Riemann Hypothesis

We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for $ζ_K(s)$. New elegant transformations are obtained when $K$ is a quadratic extension, one of which involves the modified Bessel function of the second kind.

Recursion Rules for the Hypergeometric Zeta Functions

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpre…

A note on the gaps between zeros of Epstein's zeta-functions on the critical line

It is proved that Epstein's zeta-function $ζ_{Q}(s)$, related to a positive definite integral binary quadratic form, has a zero $1/2 + iγ$ with $ T \leq γ\leq T + T^{{3/7} +\varepsilon} $ for sufficiently large positive numbers $T$. This is an improvement of the result by M. Jutila and K. Srinivas (Bull. London Math. Soc. 37 (2005) 45--53).

Multiple Dedekind Zeta Functions

In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are int…

西格尔zeta函数的模性

The modularity of Siegel's zeta functions

本文利用新近的Maass形式Weil型逆定理证明了西格尔不定二次型zeta函数的模性，完成了西格尔1938年提出的研究设想，同时证明这类zeta函数的一半对应全纯模形式。

A Kronecker limit formula for indefinite zeta functions

We prove an analogue of Kronecker's second limit formula for a continuous family of "indefinite zeta functions". Indefinite zeta functions were introduced in the author's previous paper as Mellin transforms of indefinite theta functions, as defined by Zwegers. Our formula is valid in dimension g=2 at s=1 or s=0. For a choice of parameters obeying a certain symmetry, an indefinite zeta function is a differenced ray class zeta function of a real quadratic field, and its special value at $s=0$ was conjectured by Stark to be a logarithm of an algebraic unit. Our formula also permits practical high…

Explicit evaluation of quadratic Euler sums

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.

On the zeros of Epstein zeta functions near the critical line

Let $Q$ be a positive definite quadratic form with integral coefficients and let $E(s,Q)$ be the Epstein zeta function associated with $Q$. Assume that the class number of $Q$ is bigger than $1$. Then we estimate the number of zeros of $E(s,Q)$ in the region $ \Re s > σ_T ( θ) := 1/2 + ( \log T)^{- θ}$ and $ T < \Im s < 2T$, to provide its asymptotic formula for fixed $ 0 < θ< 1$ conditionally. Moreover, it is unconditional if the class number of $Q$ is $2$ or $3$ and $ 0 < θ< 1/13$.

Secant Zeta Functions

We study the series $ψ_s(z):=\sum_{n=1}^{\infty} \sec(nπz)n^{-s}$, and prove that it converges under mild restrictions on $z$ and $s$. The function possesses a modular transformation property, which allows us to evaluate $ψ_{s}(z)$ explicitly at certain quadratic irrational values of $z$. This supports our conjecture that $π^{-k} ψ_{k}(\sqrt{j})\in\mathbb{Q}$ whenever $k$ and $j$ are positive integers with $k$ even. We conclude with some speculations on Bernoulli numbers.

斐波那契泽塔函数与延拓

The Fibonacci Zeta Function and Continuation

本文引入一类推广普通斐波那契泽塔函数的狄利克雷级数，给出将其亚纯延拓至复数域的三种方法，包括两种经典解析方法与一种基于移位卷积和模形式的方法。

zeta 函数对数导数的二次富集

Quadratic enrichment of the logarithmic derivative of the zeta function

本文将有限域上代数簇的 zeta 函数对数导数富集到 Grothendieck–Witt 群系数的幂级数，证明其与实点拓扑的联系，并对胞腔概形建立有理性结果，验证其为母题测度。

Pseudomoments of the Riemann zeta-function and pseudomagic squares

We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arithmetic factor and a geometric factor. The geometric factor is equal to the volume of the convex polytope of substochastic matrices and is equal to the leading coefficient in the expression for moments of truncated characteristic polynomial of a random unitary matrix.

Evaluations of some quadratic Euler sums

This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.

Fast methods to compute the Riemann zeta function

The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Schönhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this paper, three new fast and potentially practical methods to compute zeta are presented. One method is very simple. Its complexity has exponent 2/5. A second method relies on this author's algorithm to compute quadratic exponential sums. Its complexity has exponent 1/3. The third method employs an algorithm, developed in this p…

The Shintani double zeta functions

In this paper, we give an explicit formula of the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functions in addition to Shintani's functional equations. Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur-Selberg trace for…

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。

On the Shintani zeta function for the space of pairs of binary Hermitian forms

Throughout this paper, k is a number field. We fix a quadratic extension k_1=k(a_0) of k, where a_0=sqrt(B_0) for a certain B_0 in k^\times/(k^\times)^2. In this paper, we consider the zeta function defined for the space of pairs of binary Hermitian forms. This is the prehomogeneous vector space we discussed in section 2 [1] and is a non-split form of the D_4 case in [4]. The purpose of this paper is to determine the principal part of the adjusted zeta function. Our main result is Theorem (8.15). Our case resembles the space of pairs of binary quadratic forms which we discussed in Chapter 5 [5…