Ian Petrow, Matthew P. Young
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.
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Non-vanishing of Dirichlet $L$-functions with smooth conductors
针对大的无平方因子光滑导子，研究证明至少35.9%的本原狄利克雷L函数中心值非零，推进了狄利克雷L函数非零性的定量结果。
The Mixed Second Moment of Quadratic Dirichlet $L$-functions with Prime Conductors
本文在函数域语境下，针对首一不可约多项式平均的情况，建立了含完备L函数二阶导数的混合二阶矩的渐近公式。
In this paper we extend the hybrid Euler-Hadamard product model for quadratic Dirichlet $L$-functions associated to irreducible polynomials over function fields. We also establish an asymptotic formula for the first twisted moment in this family of L-functions and then we provide further evidence for the conjectural asymptotic formulas for its moments.
Mean square of inverses of Dirichlet $L$-functions involving conductors
本文研究Dirichlet $L$-函数的负平方矩，得到涉及导子的负二阶矩渐近公式，并应用于改进分圆数域中恢复短生成元算法的成功概率下界。
We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of $L$-functions associated to quadratic twists of a fixed elliptic curve over $\mathbb{F}_q(t)$ by monic irreducible polynomials, which allows us to show that there exists a monic irreducible polynomial such that the analytic rank of the corresponding twisted elliptic curve is…
We prove that more than nine percent of the central values $L(\frac{1}{2},χ_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $χ_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},χ_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.
Extreme central values of quadratic Dirichlet $L$-functions with prime conductors
研究素数p≡1 mod 8时L(1/2,χ_p)的下界结果，采用分析方法证明极值下限。
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.
Let $F(s)=\sum_n a_n/λ_n^s$ be a general Dirichlet series which is absolutely convergent on $\Re(s)>1$. Assume that $F(s)$ has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree $d_F$ and conductor $α_F$. In this paper, we show that there are at most $2d_F$ general Dirichlet series with a given degree $d_F$, conductor $α_F$ and residue $ρ_F$ at $s=1$. As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at $s=1$.
In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.
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.
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.
In a previous paper we proved that if an $L$-function $F$ from the Selberg class has degree $2$, its conductor $q_F$ is a prime number and $F$ is weakly twist-regular at all primes $p\neq q_F$, then $F$ has a polynomial Euler product. In this paper we extend this result to $L$-functions of degree 2 with square-free conductor $q_F$, which are weakly twist-regular at all primes $p\nmid q_F$
We establish an analogue of a conjecture of Balasubramanian, Conrey, and Heath-Brown for the family of all Dirichlet characters with conductor up to $Q$. This forms another application of our work in developing an asymptotic large sieve.
Yıldırım has classified zeros of the derivatives of Dirichlet $L$-functions into trivial zeros, nontrivial zeros and vagrant zeros. In this paper we remove the possibility of vagrant zeros for $L'(s,χ)$ when the conductors are large to some extent. Then we improve asymptotic formulas for the number of zeros of $L'(s,χ)$ in $\{s\in\mathbb{C}:\operatorname{Re}(s)>0, |\operatorname{Im}(s)|\leq T\}$. We also establish analogues of Speiser's theorem, which characterize the generalized Riemann hypothesis for $L(s,χ)$ in terms of zeros of $L'(s,χ)$, when the conductor is large.
In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)\neq 0$. To prove this, we establish asymptotic formulas of $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)$ over the family of even primitive Dirichlet characters $χ$ of conductor $q$ for more general $q$.
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.
We show that for a positive proportion of fundamental discriminants d, L(1/2,chi_d) != 0. Here chi_d is the primitive quadratic Dirichlet character of conductor d.
An algorithm is given to efficiently compute $L$-functions with large conductor in a restricted range of the critical strip. Examples are included for about 21000 dihedral Galois representations with conductor near $10^7$. The data shows good agreement with a symplectic random matrix model. Also included are an elliptic curve $L$-function and one for a quadratic Dirichlet character.
We prove a lower bound of the correct order of magnitude in the conductor aspect for small rational moments of Drichlet $L$-functions. Such bounds require new techniques, which is visible from the relationship to non-vanishing results for $L(1/2, χ)$.
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 $s=1/2$.
We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes χ)$ and $L(\tfrac{1}{2}, χ)$ do not vanish. We expect the same method to work for any large integer $q$.
Two properties of symmetric cube transfers of modular forms
本文研究对称立方转移的自守表示的导数和局部自守表示类型，通过ε因子的变化来分类可能的转移类型，重点分析p=3的情况。
$L$-functions of degree $2$ and conductor $1$: underlying ideas and a generalisation
本文简述了扩展Selberg类中次数为2且导子为1的$L$-函数分类定理，提出了涉及两个Dirichlet级数的函数方程的新推广，并纠正了原证明的疏漏。
Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples
本文用多元狄利克雷级数预测四类$L$-函数族的移位矩渐近公式，结果与CFKRS配方一致（除椭圆曲线二次扭动族需修正），并建立留数与配方项的一一对应。
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.
We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series ass…
Non-Vanishing of Dirichlet $L$-functions at the central point with restricted root number
研究通过缩小根数角限制，证明Dirichlet L-函数子族的 mollified 首二次矩渐进行为，进而证明当q趋于无穷时，偶本原字符的中心值L(1/2,χ)有正比例不消失。
Let $τ$ be the primitive Dirichlet character of conductor 4, let $χ$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $τ$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $χ\cdot τ^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space…
We consider Bernoulli distributions and their regularizations, which are measures on the $p$-adic integers $\mathbb{Z}_p$. It is well known that their Mellin transform can be used to define $p$-adic $L$-functions. We show that for $p>2$ one of the regularized Bernoulli distributions is particularly simple and equal to a measure on $\mathbb{Z}_p$ that takes the values $\pm \frac{1}{2}$ on clopen balls. We apply this to $p$-adic $L$-functions for Dirichlet characters of $p$-power conductor and obtain Dirichlet series expansions similar to the complex case. Such expansions were studied by D. Delb…
Menon's identity is a classical identity involving gcd sums and the Euler totient function $ϕ$. In a recent paper, Zhao and Cao derived the Menon-type identity $\sum\limits_{\substack{k=1}}^{n}(k-1,n)χ(k) = ϕ(n)τ(\frac{n}{d})$, where $χ$ is a Dirichlet character mod $n$ with conductor $d$. We derive an identity similar to this replacing gcd with a generalization it. We also show that some of the arguments used in the derivation of Zhao-Cao identity can be improved if one uses the method we employ here.
Zeros of Dirichlet $L$-functions on the critical line
该研究利用 Feng 的平滑因子与 Dirichlet $L$-函数均方渐近公式，证明本原特征和导子平均下至少 61.07% 的零点在临界线上、60.44% 为临界线上的单零点，改进了已有结果。
We study a double Dirichlet series of the form $\sum_{d}L(s,χ_{d}χ)χ'(d)d^{-w}$, where $χ$ and $χ'$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to $\mathbb{C}^{2}$. A convexity bound at the central point is established to be $(MN)^{3/8+\varepsilon}$ and a subconvexity bound of $(MN(M+N))^{1/6+\varepsilon}$ is proven. The developed theory is used to prove an upper bound for the smallest positive integer $d$ such that $L(1/2,χ_{dN})$ does not vanis…
Using Shimura's reciprocity law, we calculate the Galois action on the special values of theta functions associated to primitive Dirichlet characters of odd conductor, normalised by the Dedekind eta function, at the point i. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime conductor.
We study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to $\infty$. For an arbitrary positive integer $k$, we approximate the twisted $2k$th moment of this family by using Dirichlet polynomial approximations of $L^k(s,χ)$ of length $X$, with $Q<X<Q^2$. Assuming the Generalized Lindelöf Hypothesis, we prove an asymptotic formula for these approximations of the twisted moments. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith for this family of $L$-functions, and provides t…
We study $p$-adic $L$-functions $L_p(s,χ)$ for Dirichlet characters $χ$. We show that $L_p(s,χ)$ has a Dirichlet series expansion for each regularization parameter $c$ that is prime to $p$ and the conductor of $χ$. The expansion is proved by transforming a known formula for $p$-adic $L$-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored off in a natural manner from the $p$-adic Dirichlet series. We also provide an alternative proof of the expansion using $p$-adic measures and give an explicit formula for the values of the regularized Bernoulli …
A Conditional Refinement of Page's Theorem on zeros of Dirichlet $L$-functions
研究狄利克雷\(L\)函数兰道 - 西格尔零点问题，在狄利克雷\(L\)函数非实零点位于\(s = 1\)收缩邻域外的假设下，对佩奇定理进行显著细化。
Let $χ$ be a primitive Dirichlet character of conductor $q$ and let us denote by $L(z, χ)$ the associated $L$-series. In this paper, we provide an explicit upper bound for $\left|L(1, χ)\right|$ when $χ$ is a primitive even Dirichlet character with $χ(2)=1$.
Structure of relative genus fields of cubic Kummer extensions
本文研究了分圆域上三次 Kummer 扩张的相对亏格域结构，确定了使亏格群为初等双循环群的所有正整数根 $D$ 与导子 $f$。
This paper develops an analytic theory of Dirichlet series in several complex variables which possess sufficiently many functional equations. In the first two sections it is shown how straightforward conjectures about the meromorphic continuation and polar divisors of certain such series imply, as a consequence, precise asymptotics (previously conjectured via random matrix theory) for moments of zeta functions and quadratic L-series. As an application of the theory, in a third section, we obtain the current best known error term for mean values of cubes of central values of Dirichlet L-series.…
A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of nonvanishing within substantially shorter orbits generated by interm…
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…
Let $χ$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, χ)$ denote its associated Dirichlet $L$-function. We show that $|L(1, χ)| \geq 1/(9.12255 \log(q/π))$ for sufficiently large $q$, and that $|L(1, χ)| \geq 1/(9.69030 \log(q/π))$ for all $q\geq2$, improving some results of Louboutin. The improvements stem principally from the construction, via simulated annealing, of some real trigonometric polynomials having particularly favorable properties.
Firstly we prove that the Voronoi formula of Miller-Schmid type applies to automorphic forms on GL(3) for the congruence subgroup $Γ_0(N)$, when the conductor of the additive character in the formula is a multiple of $N$. As an application, we produce a result about the functional equation of $L$-function of the automorphic form on GL(3) twisted by Dirichlet characters. Secondly we prove that a similar formula applies to automorphic forms on GL(3) for the congruence subgroup $Γ_0(N)$, when the conductor of the additive character in the formula is coprime with $N$.
Let $χ$ be a primitive Dirichlet character of conductor $q$ and $L(z,χ)$ the associated L-series. In this paper we provide an explicit upper bound for $|L(1, χ)|$ when 3 divides $q$.
A note on large values of Dirichlet $L$-functions for characters of fixed order at $1/2<σ\leq 1$
本文通过简单论证，证明了固定阶本原特征的 Dirichlet $L$-函数在 $σ\in (1/2,1]$ 处存在推测性尖锐大小的大值，并给出了显式常数。
Bounds for double $L$-functions
本文研究双重 $L$-函数的阶估计问题，通过推广 Dirichlet $L$-函数到双变量情形，给出了在导子方面明确的上界。
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…
Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor
研究D4四次函数域的Artin L-函数低频零点，证明按判别式排序时至少77%的L-函数在中心点非消失，扩展了Durlanik在有理数域上的结果。
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…
We prove an asymptotic formula for the mean-square average of $L$- functions associated to subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\cal A}(p,d)$ recently introduced by E. Elma. We obtain an asymptotic formula for ${\cal A}(p,d)$ which holds true for any divisor $d$ of $p-1$ removing previous restrictions on the size of $d$. This anwers a question raised in Elma's paper. Our proof relies both on estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application we dedu…
On extreme values of quadratic twists of Dirichlet-type $L$-functions
本文将尖形式 L-函数中心点大值结论推广到 Dirichlet 型 L-函数的二次扭曲，证明对任意 ε>0 存在大量基本判别式 8d 使 |L(1/2, F ⊗ χ_{8d})| 取大值。
We study the behaviour of the dynamical zeta function and the orbit Dirichlet series for products of maps. The behaviour under products of the radius of convergence for the zeta function, and the abscissa of convergence for the orbit Dirichlet series, are discussed. The orbit Dirichlet series of the cartesian cube of a map with one orbit of each length is shown to have a natural boundary.
Sums of three cubes over a function field
本文通过函数域版本的圆方法，在假设比率猜想和特征大于3的条件下，证明了函数域F_q[t]中正比例元素可表示为三次立方的最小度数和，并探讨了二次狄利克雷L函数的类似猜想。
We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GL(n) x GL(n'). Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of $L(s,f x f~), on the residue at s=1. As an application we show that a cuspidal automorphic representation on GL(n) is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.
Comparing zeros of distinct Dirichlet L-functions
本文证明不同本原 Dirichlet 特征对应的 L 函数在指定临界带短区间内不共享零点，并将无立方因子模数情形的指数阈值改进到 1/4。
Counting Zeros of Dirichlet $L$-Functions
本文研究了Dirichlet L-函数零点的计数问题，给出了关于零点数的显式上下界，并在σ<-1/2时提供了L(s,χ)的新界。
Let $f$ be a Hecke-Maass cusp form for $SL_3(\mathbb{Z})$ and $χ$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^κ$ with $p$ prime and $κ\geq 10$. We prove a subconvexity bound $$ L\left(\frac{1}{2},π\otimes χ\right)\ll_{p,π,\varepsilon} \mathfrak{q}^{3/4-3/40+\varepsilon} $$ for any $\varepsilon>0$, where the dependence of the implied constant on $p$ is explicit and polynomial. We obtain this result by applying the circle method of Kloosterman's version, summation formulas of Poisson and Voronoi's type and a conductor lowering mechanism introduced by Munshi [14]. The…
One-level density estimates for Dirichlet L-functions with extended support
该研究无条件地将本原Dirichlet特征对应的L(s,χ)低位零点1级密度的检验函数傅里叶支撑范围扩展至[-2-50/1093,2+50/1093]，突破了此前需依赖深刻猜想的限制，并给出了对L(s,χ)非零性的相关推论。
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.
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