Weighted one-level density of low-lying zeros of Dirichlet $L$-functions

Low-lying zeros of quadratic Dirichlet $L$-functions: Lower order terms for extended support

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the Generalized Riemann Hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $ϕ$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $σ$ of the support of $\hat ϕ$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $σ=1$ involves the quantity…

关于狄利克雷L函数论的无条件估计及其在低处零点的应用

Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros

本文基于Selberg的结果，研究狄利克雷L函数的论估计，并证明了足够大的素数q对应的L函数最低非平凡零点高度小于1075·2π/log q，同时给出了首个零点位置的下界。

The n-level densities of low-lying zeros of quadratic Dirichlet L-functions

Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions where the sum of the supports of the Fourier transforms is at most 2, and showed agreement with random matrix theory predictions in this range for n < 4 but only in a restricted range for larger n. We extend these results and show agreement for n < 8, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.

One level density of low-lying zeros of families of $L$-functions

In this paper, we prove some one level density results for low-lying zeros of families of $L$-functions. More specifically, the families under consideration are that of $L$-functions of holomorphic Hecke eigenforms of level 1 and weight $k$ twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet $L$-functions.

Linear statistics of low-lying zeros of L--functions

We consider linear statistics of the scaled zeros of Dirichlet $L$--functions, and show that the first few moments converge to the Gaussian moments. The number of Gaussian moments depends on the particular statistic considered. The same phenomenon is found in Random Matrix Theory, where we consider linear statistics of scaled eigenphases for matrices in the unitary group. In that case the higher moments are no longer Gaussian. We conjecture that this also happens for Dirichlet $L$--functions.

附着于 Siegel 模形式的 L-函数的加权低矮零点

Weighted low-lying zeros of L-functions attached to Siegel modular forms

本文研究 2 次 Siegel 模形式的旋量与标准 L-函数的加权低矮零点，证明二者均具有辛对称类型，并通过按权平均扩大检验函数 Fourier 变换的支集范围，进而讨论中心值非消失性。

Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and Random Matrix Theory

The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n…

Weighted first moments of some special quadratic Dirichlet $L$-functions

In this paper, we obtain asymptotic formulas for weighted first moments of central values of families of primitive quadratic Dirichlet $L$-functions whose conductors comprise only primes that split in a given quadratic number field. We then deduce a non-vanishing result of these $L$-functions at the point $s=1/2$.

Weighted distribution of low-lying zeros of GL(2) L-functions

We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by cen…

Dirichlet $L$-函数族临界线上低零点的分布

Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions

本文通过引入高维Mellin变换，克服了短区间内Selberg磨光器计算中的交叉项障碍，证明了Dirichlet $L$-函数族在临界线上低零点数量的新下界。

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

For a totally real number field $F$ and its adèle ring $\mathbb{A}_F$, let $π$ vary in the set of irreducible cuspidal automorphic representations of ${\rm PGL}_2(\mathbb{A}_F)$ corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power $L$-functions $L(s,{\rm Sym}^r(π))$ weighted by special values of symmetric square $L$-functions $L(\frac{z+1}{2},{\rm Sym}^2(π))$ at $z \in [0, 1]$ in the level aspect. If $0 < z \le 1$, our weighted density in the level aspect has the same symmetry…

The prime number race and zeros of Dirichlet $L$-functions off the critical line. III

We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $π(x;q,a)<π(x;q,b)$ for a set of real $x$ of asymptotic density 1.

狄利克雷\(L\)函数的同时非零性，II：加权中心极限定理

Simultaneous non-vanishing of Dirichlet $L$--functions, II: Weighted central limit theorem

在广义黎曼假设下，证明四个狄利克雷\(L\)函数在中心点联合分布的加权中心极限定理，通过本原特征族扭曲，应用此定理得出该族中一定比例特征对应的中心值同时很大或同时非零且很小。

以中心L值幂为权的希尔伯特模L函数的低零点

Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values

该研究针对权为$\textbf{k}$、素水平为$\mathfrak{q}$的本原希尔伯特模形式集，分析以中心L值幂为权的L函数低零点单级密度，证明r=1,2,3时分布与随机矩阵理论预测一致，并给出一般r≥1时的猜想公式。

Weighted Average Number of Prime $m$-tuples lying on an Admissible $k$-tuple of Linear Forms

We find an upper bound for the sum $\sum_{x<n\leq 2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$, where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the admissible set $\mathcal{H}=\{h_{1},...,h_{k}\}$, $m\geq 1$ and $x$ is sufficiently large, with the same weights $w_{n}$ used in the Maynard's paper "Dense clusters of primes in subsets". The estimate will be uniform over positive integer $k$ with $m+1\leq k\leq (\log x)^{1/5}$ and on admissible set $\mathcal{H}$ with $0\leq h_{1}<...<h_{k}\leq x$. The upper bound will depend on a…

Simultaneous Diophantine approximation on affine subspaces and Dirichlet improvability

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the base point of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence. We also prove a partial analogue regarding the Hausdorff measure theory. Furthermore, we obtain various results relating weighted Diophantine approximation and Dirichlet improvability. In particu…

Maass形式L函数族中的低零点：一个扩展密度定理

Low-lying zeros in families of Maass form L-functions: an extended density theorem

本文研究素数水平Maass形式L函数族低零点密度，将可容许支撑从(-3/2,3/2)扩展到(-15/8,15/8)，并假设大密度猜想可至(-2,2)，方法基于Kutznetsov迹公式、大筛法及四次矩界。

关于扭曲周期函数与 Dirichlet L-函数在临界线上加权均方均值的注记

On twisted period functions and Moments of a weighted mean square of Dirichlet L-functions on the critical line

本文将周期函数等工具推广至任意本原特征的 Dirichlet L-函数，给出了加权均方测度矩的闭式表达式，完整刻画了临界线上的模性质，并导出了适用于所有本原特征的无限个非平凡正性条件。

Selberg's Central limit theorem for quadratic Dirichlet L-functions over function fields

In this article, we study the logarithm of the central value $L\left(\frac{1}{2}, χ_D\right)$ in the symplectic family of Dirichlet $L$-functions associated with the hyperelliptic curve of genus $δ$ over a fixed finite field $\mathbb{F}_q$ in the limit as $δ\to \infty$. Unconditionally, we show that the distribution of $\log \big|L\left(\frac{1}{2}, χ_D\right)\big|$ is asymptotically bounded above by the Gaussian distribution of mean $\frac{1}{2}\log °(D)$ and variance $\log °(D)$. Assuming a mild condition on the distribution of the low-lying zeros in this family, we obtain the full Gaussian …

$L$-函数的零点与Dirichlet系数的大偏和

Zeros of $L$-functions and large partial sums of Dirichlet coefficients

该研究针对满足弱广义Ramanujan猜想的$L$-函数，证明其Dirichlet系数的大偏和会排斥低零点远离临界线，拓展并定量改进了前人相关研究成果。

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…

关于加权复合算术函数的均值

On the Mean Value of a Weighted Composite Arithmetic Function

本文利用解析数论方法，研究了包含Dirichlet除数函数和广义最小幂函数的复合函数的均值性质，建立了加权和函数的渐近公式。

关于Dirichlet $L$-函数在正整数处乘积的均值

On the mean values of products of Dirichlet $L$-functions at positive integers

本文研究Dirichlet $L$-函数在正整数处乘积的均值分布，给出了由Dirichlet特征加权、涉及Bernoulli函数和Jordan totient函数的显式公式，推广了多种已知结果。

On the Low-lying zeros of Hasse-Weil L-functions for Elliptic Curves

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevic group. Statements of this fla…

The first moment of quadratic Dirichlet L-functions

We obtain an asymptotic formula for the smoothly weighted first moment of primitive quadratic Dirichlet L-functions at the central point, with an error term that is "square-root" of the main term. Our approach uses a recursive technique that feeds the result back into itself, successively improving the error term.

The prime number race and zeros of Dirichlet L-functions off the critical line, II

We continue our examination the effects of certain hypothetical configurations of zeros of Dirichlet $L$-functions lying off the critical line ("barriers") on the relative magnitude of the functions $π_{q,a}(x)$. Here $π_{q,a}(x)$ is the number of primes $\le x$ in the progression $a \mod q$. In particular, we construct barriers so that $π_{q,1}(x)$ is simultaneously greater than, or simultaneously less than, each of $k$ functions $π_{q,a_i}(x)$ ($1\le i\le k$). We also construct barriers so that only a small number of the $r!$ possible orderings of functions $π_{q,a_i}(x)$ ($1\le i\le r$) occ…

Zeros of Dirichlet L-functions over Function Fields

Random matrix theory has successfully modeled many systems in physics and mathematics, and often the analysis and results in one area guide development in the other. Hughes and Rudnick computed $1$-level density statistics for low-lying zeros of the family of primitive Dirichlet $L$-functions of fixed prime conductor $Q$, as $Q \to \infty$, and verified the unitary symmetry predicted by random matrix theory. We compute $1$- and $2$-level statistics of the analogous family of Dirichlet $L$-functions over $\mathbb{F}_q(T)$. Whereas the Hughes-Rudnick results were restricted by the support of the…

Siegel's mass formula and averages of Dirichlet L-functions over function fields

Let D be a square-free polynomial in F_q[t], where q is odd, and let G be a genus of definite ternary lattices over F_q[t] of determinant D. In this paper we give self-contained and relatively elementary proofs of Siegel's formulas for the weighted sum of primitive representations numbers over the classes of G and for the mass of G. Our proof of the mass formula shows an interesting relation with certain averages of Dirichlet L-functions.

Riemann zeta 函数的无零点区域与加权 Dirichlet 空间中的逼近

Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces

本文研究黎曼Zeta函数在加权Dirichlet空间中的无零点区域，证明了对标准加权空间D_α(α∈(-3,-2))的逼近问题可给出半平面无零点条件，并将结果推广至解析函数空间ℓ^p_α，特例情形给出了素数定理的新等价表述。

Weighted uniform Diophantine approximation of systems of linear forms

Following the development of weighted asymptotic approximation properties of matrices, we introduce the analogous uniform approximation properties (that is, study the improvability of Dirichlet's Theorem). An added feature is the use of general norms, rather than the supremum norm, to quantify the approximation. In terms of homogeneous dynamics, the approximation properties of an $m \times n$ matrix are governed by a trajectory in $\mathrm{SL}_{m+n}({\mathbb R})/\mathrm{SL}_{m+n}({\mathbb Z})$ avoiding a compact subset of the space of lattices called the critical locus defined with respect to …

非均匀伸缩平移下收缩光滑曲线的等分布与加权Dirichlet逼近

Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation

本文研究非均匀伸缩作用下非退化光滑曲线平移在幺模格空间中的等分布，证明固定大小及收缩曲线段均趋于等分布，并由此推出加权Dirichlet逼近定理对几乎所有点不可改进，推广了Shah的解析曲线结果。

Curves, dynamical systems and weighted point counting

Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet L-series of the two curves are equal via an isomorphism of their Dirichlet character group…