Weyl group multiple Dirichlet series constructed from quadratic characters

Weyl group multiple Dirichlet series of type A_2

A Weyl group multiple Dirichlet series is a Dirichlet series in several complex variables attached to a root system Phi. The number of variables equals the rank r of the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group W of Phi. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n-th order Gauss sums attached to the root system of type A_2. The basic technique is to construct a rational function in r variables invariant under a certain action of W, and use this to build a ``local factor'' of…

On the p-parts of quadratic Weyl group multiple Dirichlet series

Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, which has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. Two constructions of such series are available, one based on summing products of n-th order Gauss sums, the second based on averaging a certain group action over the Weyl group. In this paper we study these constructions and the relationship betwee…

Twisted Weyl group multiple Dirichlet series over the rational function field

Weyl group multiple Dirichlet series are Dirichlet series in $r$ complex variables, with analytic continuation to $\mathbb{C}^r$ and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank $r$. Such series may be defined for any global field $K$, but in the case when $K$ is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when $K=\mathbb{F}_q(T)$, and describe the denominators and support of the numerators.

The Weyl bound for Dirichlet $L$-functions of cube-free conductor

We prove a Weyl-exponent subconvex bound for any Dirichlet $L$-function of cube-free conductor. We also show a bound of the same strength for certain $L$-functions of self-dual $\mathrm{GL}_2$ automorphic forms that arise as twists of forms of smaller conductor.

The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound

We prove a Lindelöf-on-average upper bound for the fourth moment of Dirichlet $L$-functions of conductor $q$ along a coset of the subgroup of characters modulo $d$ when $q^*|d$, where $q^*$ is the least positive integer such that $q^2|(q^*)^3$. As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet $L$-functions with no restrictions on the conductor.

Affine Weyl Group Multiple Dirichlet Series: Type $\tilde{A}$

We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac-Moody root system $\tilde{A}_n$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is over the rational function field $\mathbb{F}_q(t)$, and is based upon four natural axioms from algebraic geometry. We prove that the four axioms yield a unique series with meromorphic continuation to the largest possible domain and the desired infinite group of symmetries.

Bhargava Integer Cubes and Weyl Group Multiple Dirichlet Series

We investigate the Shintani zeta functions associated to the prehomogeneous spaces, the example under consideration is the set of $2 \times 2\times 2$ integer cubes. We show that there are three relative invariants under a certain parabolic group action, they all have arithmetic nature and completely determine the equivalence classes. We show that the associated Shintani zeta function coincides with the $A_3$ Weyl group multiple Dirichlet series. Finally, we show that the set of semi-stable integer orbits maps finitely and surjectively to a certain moduli space.

Quadratic Weyl group multiple Dirichlet series of Type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$

In this paper and its sequel \cite{DPP}, we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead \cite{White2} and \cite{White1}. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this…

axiomatic multiple Dirichlet series 的函数方程，Weyl groupoids 和量子代数

Functional equations of axiomatic multiple Dirichlet series, Weyl groupoids, and quantum algebra

本文通过五条几何公理定义多重Dirichlet级数并推导其函数方程，将其与Weyl群oid关联，并应用于量子代数中的Nichols代数上同调研究。

Multiple Dirichlet Series for Affine Weyl Groups

Let $W$ be the Weyl group of a simply-laced affine Kac-Moody Lie group, excepting $\tilde{A}_n$ for $n$ even. We construct a multiple Dirichlet series $Z(x_1, \ldots x_{n+1})$, meromorphic in a half-space, satisfying a group $W$ of functional equations. This series is analogous to the multiple Dirichlet series for classical Weyl groups constructed by Brubaker-Bump-Friedberg, Chinta-Gunnells, and others. It is completely characterized by four natural axioms concerning its coefficients, axioms which come from the geometry of parameter spaces of hyperelliptic curves. The series constructed this w…

Sub-Weyl subconvexity for Dirichlet L-functions to prime power moduli

We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving p-adically analytic phases, which can be naturally seen as a p-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including…

二次 Weyl 群多重 Dirichlet 级数的留数

Residues of quadratic Weyl group multiple Dirichlet series

本文研究有限不可约根系的 Chinta-Gunnells 平均在单短根极除子处的留数，给出由正交根子系平均表示的显式公式，并推广到有理函数域与 Gauss 域上的二次 Weyl 群多重 Dirichlet 级数。

可对称化卡-穆迪根系的外尔群多重狄利克雷级数的一种分解

A decomposition of Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems

研究可对称化卡-穆迪根系的扭曲外尔群多重狄利克雷级数，用钦塔-冈内尔斯方法构建\(p\)部分，给出函数分解定理，证明其对有理函数域上的级数成立，还得到仿射\(\widetilde{A}_1\)情形下的额外函数方程及显式公式。

元辛覆盖上的爱森斯坦级数与多重狄利克雷级数

Eisenstein Series on Metaplectic Covers and Multiple Dirichlet Series

本文计算了整体元辛群上由环面诱导的爱森斯坦级数的首个惠特克系数，并将其与对偶根系统的外尔群多重狄利克雷级数建立联系，从而证实了 Brubaker-Bump-Friedberg 猜想。

Weyl-type hybrid subconvexity bounds for twisted $L$-functions and Heegner points on shrinking sets

We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.

关于与固定阶Dirichlet特征相关的$L$-函数零点的注记

A note on the zeros of $L$-functions associated to fixed-order Dirichlet characters

该研究利用Dirichlet $L$-函数的Weyl界推导固定阶Dirichlet特征族对应$L$-函数的零点密度估计，在σ距临界线足够远时改进了作者此前的界。

Crystal constructions in Number Theory

Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics.

Shifted Multiple Dirichlet Series

We develop certain aspects of the theory of shifted multiple Dirichlet series and study their meromorphic continuations. These continuations are used to obtain explicit spectral first and second moments of Rankin-Selberg convolutions. One consequence is a Weyl type estimate for the Rankin-Selberg convolution of a holomorphic cusp form and a Maass form with spectral parameter $|t_j|\le T$, namely: $$ \left| L\left(\frac{1}{2}+ir, f\times u_j\right) \right| \ll_N T^{2/3+ε}, $$ uniformly, for $|r| \le T^{2/3}$, with the implied constant depending only on $f$ and the level $N$.

A Weyl Creation Algebra Approach to the Riemann Hypothesis

We sketch a Weyl creation operator approach to the Riemann Hypothesis; i.e.,arithmetic on the Weyl algebras with ergodic theory to transport operators. We prove that finite Hasse-Dirichlet alternating zeta functions or eta functions can be induced from a product of "creation operators". The latter idea is the result of considering complex quantum mechanics with complex time-space related concepts. Then we overview these ideas, with variations, which may provide a pathway that may settle RH; e.g., a Euler Factor analysis to study the quantum behavior of the integers as one pushes up the critica…

与二次曲面相关的多重 Dirichlet 级数

Multiple Dirichlet series associated with quadrics

本文定义了与二次曲面相关的多重 Dirichlet 级数，通过构造二次空间滤过、定义抛物群作用并应用非交换 Poisson 求和公式获得函数方程，并证明其函数方程群同构于 A3 型有限 Weyl 群。

Weyl-type bounds for twisted $GL(2)$ short character sums

Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients $λ_{f}(n)$. Let $χ$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the following Weyl-type bound: for any $ε>0$, $$\sum_{|n| \ll N}λ_{f}(n)χ(n) \ll_{f,ε}N^{3/4 }p^{1/6}(pN)^ε.$$ We can see an improvement of the range $N > p^{3/4}$ to the range $N > p^{2/3}$ and we get a bound for $S_{f,χ}(N)$ without going into the $L$-function.

Second moments of Rankin-Selberg convolution and shifted Dirichlet series

In this paper we work over $Γ_0(N)$, for any $N$ and write the spectral moment of a product of two distinct Rankin-Selberg convolutions at a general point on the critical line $\frac{1}{2}+it$ as a main term plus a sharp error term in the $t$ aspect and the spectral aspect. As a result we obtain hybrid Weyl type subconvexity results in the $t$ and spectral aspects. Also, for fixed modular forms $f$, $g$ of even weight $k\geq 4$ we show there exists a Maass cusp form $u_j$ such that $L(1/2, f\times u_j)$, $L(1/2, g\times u_j)$ are simultaneous non-zero.

An explicit hybrid estimate for $L(1/2+it,χ)$

An explicit hybrid estimate for $L(1/2+it,χ)$ is derived, where $χ$ is a Dirichlet character modulo $q$. The estimate applies when $t$ is bounded away from zero, and is most effective when $q$ is powerfull, yielding an explicit Weyl bound in this case. The estimate takes a particularly simple form if $q$ is a sixth power. Several hybrid lemmas of van der Corput--Weyl type are presented.

Multiple Dirichlet series associated to additive and Dirichlet characters

In this article we study analytic properties of the multiple Dirichlet series associated to additive and Dirichlet characters. For the multiple Dirichlet series associated to additive characters, the meromorphic continuation is established via obtaining translation formulas satisfied by these multiple Dirichlet series. While it seems difficult to obtain such a translation formula for the multiple Dirichlet series associated to Dirichlet characters, we rely on their intrinsic connection with the multiple Dirichlet series associated to additive characters in order to investigate their analytic c…

Multiple Lucas Dirichlet series associated to additive and Dirichlet characters

In this article, we obtain the analytic continuation of the multiple shifted Lucas zeta function, multiple Lucas $L$-function associated to Dirichlet characters and additive characters. We then compute a complete list of exact singularities and residues of these functions at these poles. Further, we show the rationality of the multiple Lucas $L$-functions associated with quadratic characters at negative integer arguments.

Sums with convolution of Dirichlet characters

We improve a recent result by Shparlinski and Banks related to sums with convolution of Dirichlet characters.

Multiplicative Functions and a Taxonomy of Dirichlet Characters

A method of estimating sums of multiplicative functions braided with Dirichlet characters is demonstrated, leading to a taxonomy of the characters for which such sums are large.

Twists by Dirichlet characters and polynomial Euler products of L-functions

We prove that suitable properties of the twists by Dirichlet characters of an L-function of degree 2 imply that its Euler product is of polynomial type.

Mean values of cubic and quartic Dirichlet Characters

We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet character sums.

Character Average of Second and Fourth Powers of Dirichlet L-Series at Unity

We obtain by simple elementary method, the character average of fourth power of Dirichlet L-series at unity.This result seems to be new.We also obtain character average of second power of Dirichlet L-series at unity,using the power series in second variable of Hurwitz zeta function at unity.

The equidistribution of L-functions of twists by Witt vector Dirichlet characters over function fields

Katz showed that the L-functions of all Dirichlet characters of F_q(t), with conductor a fixed power of a degree one prime, are equidistributed in the limit as q goes to infinity. We generalize this statement to the L-functions of twists of an arbitrary Galois representation by Dirichlet characters, including independence of the L-functions of twists of different representations by the same Dirichlet character. A similar generalization, without the independence statement, for characters with squarefree conductor, was proven by Hall, Keating, and Roditty-Gershon.

函数方程刻画狄利克雷特征

Functional Equations Characterize Dirichlet Characters

本文证明，在温和假设下，形如∑_{n≥1} f(n)n^{-s}的L级数满足的函数方程迫使系数函数f为原初狄利克雷特征，从而也迫使欧拉乘积存在。

素数模二次Dirichlet特征和矩的界

Bounds for moments of quadratic Dirichlet character sums of prime moduli

本文在广义黎曼假设下，先对模8p的二次Dirichlet L-函数平移矩建立精确上界，再据此证明素数模二次Dirichlet特征和矩的界。

Computing Dirichlet character sums to a power-full modulus

The Postnikov character formula is used to express large portions of a Dirichlet character sum in terms of quadratic exponential sums. The quadratic sums are then computed using an analytic algorithm previously derived by the author. This leads to a power-saving if the modulus is smooth enough. As an application, a fast, and potentially practical, method to compute Dirichlet L-functions with complexity exponent 1/3 for smooth enough moduli is derived.

狄利克雷特征群生成元上狄利克雷L函数的一阶矩

The First Moment of Dirichlet L-functions over Generators of the Character Group

研究当q通过素数趋于无穷时，模q的狄利克雷特征群生成元上狄利克雷L函数中心值L(1/2,χ)的平均值，给出相应评估。

二次 Dirichlet 特征和的矩界

Bounds for moments of quadratic Dirichlet character sums

本文研究平滑二次 Dirichlet 特征和的矩估计问题，在广义 Riemann 假设下证明其上界，从而确认 M. Jutila 的相关猜想。

Estimates for character sums and Dirichlet $L$-functions to smooth moduli

We use the $q$-analogue of van der Corput's method to estimate short character sums to smooth moduli. If $χ$ is a primitive Dirichlet character modulo a squarefree, $q^δ$-smooth integer $q$ we show that $$L(\frac12,χ)\ll_εq^{\frac{27}{164}+O(δ)+ε}.$$

实狄利克雷特征的比率猜想与多重狄利克雷级数

The Ratios conjecture for real Dirichlet characters and multiple Dirichlet series

本文在广义黎曼假设下，利用多重狄利克雷级数证明实狄利克雷特征族的单平移L函数比率猜想，扩展至非本原特征可增大平移范围，进而得到平移对数导数和的渐近公式。

关于无立方Hecke特征的Weyl次凸界：全实情形

On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case

本文证明了全实域上无立方级Hecke特征的Weyl型次凸界，通过显式反演Motohashi公式并利用Schwartz函数与分布不变性。

关于Dirichlet特征上L-函数的矩

On moments of L-functions over Dirichlet characters

本文为Dirichlet L-函数二阶矩的Heath-Brown完整渐近展开提供了新证明，并将相应渐近展开推广到Hecke-Maass L-函数的扭曲一阶矩。

A generalization of Tóth identity in $\mathbb{F}_q[T]$ involving a Dirichlet Character

Let $\mathbb{A}=\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$. In this article, we prove a generalization of Tóth identity on $\mathbb{A}$ involving arithmetical functions, multiplicative and additive characters.

Infinite products involving Dirichlet characters and cyclotomic polynomials

Using some basic properties of the gamma function, we evaluate a simple class of infinite products involving Dirichlet characters as a finite product of gamma functions and, in the case of odd characters, as a finite product of sines. As a consequence we obtain evaluations of certain multiple $L$-series. In the final part of this paper we derive expressions for infinite products of cyclotomic polynomials, again as finite products of gamma or of sine functions.

二次狄利克雷特征和的矩

Moments of quadratic Dirichlet character sums

本文研究二次狄利克雷特征和的高次幂矩，利用多变量 Tauberian 定理得到渐近行为，并给出 Jutila 猜想中对数因子指数的下界及 L-函数平移矩加权平均的下界。

Asymptotics of generalized partial theta functions with a Dirichlet character

In this paper, we prove asymptotic expansions of generalized partial theta functions with a nonprincipal Dirichlet character and relate these expansions to certain $L$-series.

Central Limit Theorems for series of Dirichlet characters

For a given Dirichlet character $χ(n) = e^{i θ_n}$, we prove central limit theorems for the series $\sum_{p'} \cos θ_{p'}$ for non-principal characters, and $\sum_{p' } \cos (t \log p')$ for principal characters, where $p'$ are integers based on a variant of Cramér's random model for the primes. For non-principal characters, we use these results to show that the Generalized Riemann Hypothesis for the associated $L$-function is true with probability equal to one. For principal characters we propose how to extend these arguments to $\Re (s) = t \to \infty$.

Menon-type identities concerning Dirichlet characters

Let $χ$ be a Dirichlet character (mod $n$) with conductor $d$. In a quite recent paper Zhao and Cao deduced the identity $\sum_{k=1}^n (k-1,n) χ(k)= φ(n)τ(n/d)$, which reduces to Menon's identity if $χ$ is the principal character (mod $n$). We generalize the above identity by considering even functions (mod $n$), and offer an alternative approach to proof. We also obtain certain related formulas concerning Ramanujan sums.

On a class of elliptic functions associated with the even Dirichlet characters

We construct a class of companion elliptic functions associated with the even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $\wp(z|τ)$ as the blueprint, we will derive their representations in terms of $q$-series and partial fractions, explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups $Γ_0(N)$ and $Γ_1(N)$.

Dirichlet 特征的群聚

Murmurations of Dirichlet characters

本文研究两族 Dirichlet 特征的群聚密度，分别给出与实验相符的群聚函数及类似 Zubrilina 密度的普适性，并解释其与辛族 L 函数 1 层密度相变的联系。

On Primitivity and Vanishing of Dirichlet Series

For a rational valued periodic function, we associate a Dirichlet series and provide a new necessary and sufficient condition for the vanishing of this Dirichlet series specialized at positive integers. This question was initiated by Chowla, and carried out by Okada for a particular infinite sum. Our approach relies on the decomposition of the Dirichlet characters in terms of primitive characters. Using this, we find some new family of natural numbers for which a conjecture of Erdös holds. We also characterize rational valued periodic functions for which the associated Dirichlet series vanishe…

A fast amortized algorithm for computing quadratic Dirichlet $L$-functions

An algorithm to compute Dirichlet $L$-functions for many quadratic characters is derived. The algorithm is optimal (up to logarithmic factors) provided that the conductors of the characters under consideration span a dyadic window.

具有固定阶特征的某些本原 Dirichlet L-函数中心值的一阶矩

First moment of central values of some primitive Dirichlet $L$-functions with fixed order characters

本文利用二重 Dirichlet 级数方法渐近计算本原三次、四次和六次 Dirichlet L-函数族中心值的平滑一阶矩，并证明了这些 L-值的定量非零结果。

关于例外特征与 Dirichlet $L$-函数非零性的一个注记

A note on exceptional characters and non-vanishing of Dirichlet $L$-functions

本文在存在例外实本原 Dirichlet 特征的假设下，证明大范围素数模上几乎所有 Dirichlet 特征的 $L$-函数在中心点非零。

GL(2) L-函数在狄利克雷字符扭曲下的互易性

Reciprocity for GL(2) L-functions twisted by Dirichlet characters

本文通过新方法建立了GL(2) L-函数在狄利克雷字符扭曲下的第一矩互易关系，扩展了先前研究成果。

Determinants related to Dirichlet characters modulo 2, 4 and 8 of binomial coefficients and the algebra of recurrence matrices

Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8.