Djordje Milićević, Daniel White
We prove the q-aspect analogue of Heath-Brown's result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. Our results rely on the p-adic method of stationary phase for sums of products and complement Nunes' bound for smooth square-free moduli.
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We prove an analogue of Heath-Brown's bound on the twelfth moment of the Riemann zeta function for Dirichlet L-functions with smooth moduli.
More than five-twelfths of the zeros of $ζ$ are on the critical line
本文利用 $ζ$ 比率自相关技术无条件计算了扭曲黎曼 zeta 函数的第二矩，借此描述高阶导数线性组合的平滑处理过程，并改进长平滑子与 Kloosterman 和的结果，将临界零点比例下界提升至略高于十二分之五。
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。
Large values of quadratic Dirichlet $L$-functions of prime-related moduli
本文在广义黎曼假设下，应用长谐振子方法，展示了与素数相关模数的二次狄利克雷 $L$-函数族在中心点处的大值。
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 $$L(\frac12,χ)\ll_εq^{\frac{27}{164}+O(δ)+ε}.$$
The Dirichlet characters of reduced residue systems modulo m are tabulated for moduli m <= 195. The associated L-series are tabulated for m <= 14 and small positive integer argument s accurate to 10^(-50), their first derivatives for m <= 6. Restricted summation over primes only defines Dirichlet Prime L-functions which lead to Euler products (Prime Zeta Modulo functions). Both are materialized over similar ranges of moduli and arguments. Formulas and numerical techniques are well known; the aim is to provide direct access to reference values.
Large values of cubic and quartic Dirichlet $L$-functions of prime moduli
本文在广义黎曼假设下，应用谐振子方法研究素数模的三次与四次Dirichlet L-函数族，展示其在中心点处的大值。
Extreme values of quadratic Dirichlet $L$-functions with prime-related moduli
研究具有素数相关模的二次狄利克雷\(L\)函数极值，通过给出新下界推广前人工作并改进近期成果。
In this paper, we study the $k$-th moment of central values of the family of quadratic Dirichlet $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generlized Riemann hypothesis, we establish sharp upper and lower bounds for the $k$-th power moment of these $L$-values for all real $k \geq 0$.
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…
Motohashi's Formula for the Fourth Moment of Individual Dirichlet $L$-Functions and Applications
本文建立了Dirichlet L函数的新互反公式，将单个Dirichlet L函数四阶矩与Hecke-Maass新形式中心L值三阶矩关联，推广了Iwaniec、Jutila和Heath-Brown的结果，是Petrow-Young工作的逆版本。
The mollified fourth moment of Dirichlet $L$-functions
该研究推导了带幂节省误差项的Dirichlet $L$-函数族软化四阶矩渐近公式，适用更广模集合且精度更高，改进了中心点四阶矩的已有渐近估计结果。
The asymptotic formula of the fourth moment of Dirichlet $L$-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved by M. P. Young in 2011. This work establishes this asymptotic formula for general moduli. The work relies on the study of a special divisor sum function, called $\mathcal{D}_q$-function, which plays a key role in deducing the main terms. Another key ingredient is an application of new bounds for double sums in Kloosterman sums, applied in the estimate of th…
Ratios conjecture of quadratic Hecke $L$-functions of prime-related moduli
该研究在广义黎曼假设下，针对类数为1的虚二次数域中与素数相关模的二次狄利克雷及Hecke L-函数族，建立了带移位的L-函数比率猜想，并据此渐近计算了中心值一阶矩与单级密度。
We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.
Let $M=M_1 M_2 M_3$ be the product of three distinct primes and let $χ=χ_1 χ_2 χ_3$ be a Dirichlet character of modulus $M$ such that each $χ_i$ is a primitive character modulo $M_i$ for $i=1,2,3$. In this paper, we provide a $δ$-symbol method for obtaining non-trivial cancellation in smooth character sums of the form $\sum_{n=1}^\infty χ(n) W(n/N)$, with $N$ roughly of size $\sqrt M$ and $W$ a smooth compactly supported weight function on $(0, \infty)$. As a corollary, we establish hybrid subconvexity bounds for the associated Dirichlet $L$-function.
In this paper, we establish the expected order of magnitude of the $k$th-moment of quadratic Dirichlet $L$-functions associated to hyperelliptic curves of genus $g$ as well as of prime moduli over a fixed finite field $\mathbb{F}_{q}$ for all real $k \geq 0$.
In this paper we provide an explicit construction of a $distinctive$ multiple Dirichlet series associated to products of quadratic Dirichlet L-series, which we believe should be tightly connected to a generalized metaplectic Whittaker function on the double cover of a Kac-Moody group. To do so, we first impose a set of axioms, independent of any group of functional equations, which the aforementioned object should satisfy. As a consequence, we deduce that the coefficients of the $p$-parts of the multiple Dirichlet series satisfy certain recurrence relations. These relations lead to a family of…
We study the first moments of central values of Hecke $L$-functions associated with quadratic, cubic and quartic symbols to prime moduli. This also enables us to obtain results on first moments of central values of certain families of cubic and quartic Dirichlet $L$-functions of prime moduli.
Zero-Density Theorems for Families of Dirichlet $L$-functions
本文针对与特定稀疏模数集合及固定阶特征相关的 Dirichlet $L$-函数族，证明了若干零点密度定理。
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.
The eighth moment of Dirichlet L-functions II
本研究无条件证明了模q≤Q的本原特征下、临界线短平均的Dirichlet L函数第八矩渐近公式，突破了此前该结果依赖广义黎曼假设的限制。
We prove an asymptotic for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $χ$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line, conditionally on GRH. We derive the analogous result for the fourth moment of Dirichlet twists of GL(2) L-functions. Our results match the moment conjectures in the literature; in particular, the constant 24024 appears as a factor in the leading order term of the eighth moment.
We evaluate some twisted fourth moment of Dirichlet $L$-functions at the central point s=1/2 and for prime moduli q. The principal tool is a careful analysis of a shifted convolution problem involving the divisor function using spectral theory of automorphic forms and bounds for bilinear forms in Kloosterman sums. Having in mind simultaneous non vanishing results, we apply the Theorem to establish an asymptotic formula of a mollified fourth moment for this family of $L$-functions
We prove a formula, with power savings, for the sixth moment of Dirichlet L-functions averaged over moduli $q$, over primitive characters $χ$ modulo $q$, and over the critical line. Our formula agrees precisely with predictions motivated by random matrix theory. In particular, the constant 42 appears as a factor in the leading order term, exactly as is predicted for the sixth moment of the Riemann zeta-function.
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.
Effective hybrid joint universality for Dirichlet $L$-functions and its application
本文研究了Dirichlet L-函数的混合联合普遍性定理，并应用于估计Hurwitz ζ-函数的普遍性定理的下密度下界。
Explicit Burgess inequalities for cubefree moduli
本文改进了r=2时的显式Burgess不等式常数，并为立方自由模数q的r≥3情况提供了显式界。
Central non-vanishing of Dirichlet $L$-functions
本文证明对充分大且不满足 $q\equiv2\bmod4$ 的模 $q$，其原始特征中心值非零的比例至少为 $3/8-o(1)$，并在特定算术条件下得到更强比例。
Ratios conjecture of quartic $L$-functions of prime moduli
在广义黎曼假设下，利用多重狄利克雷级数方法，研究了四次海格尔 $L$-函数在素数模上的比率猜想，并评估了中心值的一阶矩和一级密度。
Let $Δ(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|ζ(1/2 + it)|$. If $E^*(t) = E(t) - 2πΔ^*(t/(2π))$ with $Δ^*(x) = - Δ(x) +2Δ(2x)- {1\over2}Δ(4x)$, then we obtain $$ \int_0^T(E^*(t))^4 dt \ll_εT^{16/19+varepsilon}$$, which is the first non-trivial bound for higher moments of $E^*(t)$. The method of proof also provides an upper bound for sums of fourth powers of mean square integrals of $|ζ(1/2 + it)|$ over well-spaced points. This, in turn, yields a new proof of the twelfth moment estimate for $|ζ(1/2…
We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair $(ω_1,ω_2)$ of multiplicative characters modulo q, there is a positive proportion of $χ$ (mod q) such that $L(χ, 1/2 ), L(χω_1, 1/2 )$ and $L(χω_2, 1/2)$ are simultaneo…
Ratios conjecture of cubic $L$-functions of prime moduli
本文在广义黎曼假设下，针对三次 Hecke L 函数提出比率猜想，并应用该猜想计算了中心值的第一矩和一级密度。
For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $φ(q)$ reduced residue classes modulo $q$. This paper considers patterns of sequences of consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+k})$ modulo $q$. Numerical evidence suggests a preference for certain prime patterns. For example, computed frequencies of the pattern $(a,a)$ modulo $q$ up to $x$ are much less than the expected frequency $π(x)/φ(q)^2$. We begin to rigorously c…
Mixed moments of twisted $L$-functions
本文研究了狄利克雷 $L$-函数和自守 $L$-函数在所有原始特征下的扭曲混合矩，推导出具有最优误差项的渐近公式，扩展了先前关于素数模数的研究结果。
The sixth moment of Dirichlet L-functions at the central point
本文移除了Conrey、Iwaniec和Soundararajan在处理Dirichlet L-函数中心点六次矩时引入的额外短t-平均，克服了由此产生的非平衡和式困难，从而为Huxley的原始矩问题建立了渐近公式。
Under the Riemann Hypothesis for Dirichlet L-functions, we improve on the error term in a smoothed version of an estimate for the density of elliptic curves with square-free $Δ=D/16$, where D is the discriminant, by T.D. Browning and the author. To achieve this improvement, we elaborate on our methods for counting weighted solutions of inhomogeneous cubic congruences to power-ful moduli. The novelty lies in going a step further in the explicit evaluation of complete exponential sums and saving a factor by averaging over the moduli.
In this paper, we confirm a smoothed version of a recent conjecture on the variance of the k-fold divisor function in arithmetic progressions to individual composite moduli, in a restricted range. In contrast to a previous result of Rodgers and Soundararajan, we do not require averaging over the moduli. Our proof adapts a technique of S. Lester who treated in the same range the variance of the k-fold divisor function in the short intervals setting, and is based on a smoothed Voronoi summation formula but twisted by multiplicative characters. The use of Dirichlet characters allows us to extend …
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。
Oscillatory Interference in Dirichlet L-Functions and the Separation of Primes
本文通过构造Dirichlet L-函数非平凡零点的振荡重建，揭示了素数分离的机制，并展示了零点分布如何生成结构化的振荡图案。
Large Value Estimates for Dirichlet Polynomials with Characters and Zero Density of Dirichlet $L$-Functions
研究狄利克雷多项式大值估计及狄利克雷\(L\)函数零点密度，通过改编Guth--Maynard方法改进指数，得到零点密度估计，进而推导模素数算术级数中最小哥德巴赫数新上界及短区间内算术级数中素数的新结果。
Non-vanishing and One Level Density for Dirichlet $L$-functions Along Short Averages
本文研究Dirichlet L-函数中心值在短区间和算术级数中平均的非零比例，证明在短平均下可超过1/2-δ，并在广义黎曼假设下利用单层密度方法获得超过1/2的非零比例。
On the Erdős--Odlyzko--Sárközy conjecture for prime-power moduli
该研究针对素数幂模q=p^γ，证明Erdős–Odlyzko–Sárközy猜想中第三个素数可取次多项式小量，给出相关渐近公式并验证多数既约剩余类满足猜想，还提供了任意q下的条件版本。
A geometric approach to functional equations for general multiple Dirichlet series over function fields
本文针对Sawin未解决的函数域$\mathbb{F}_{q}(T)$上多重狄利克雷级数收敛性与函数方程问题，通过反常层分解定理结合Kontsevich模空间紧化证明其收敛性，利用多变量推广的狄利克雷特征-傅里叶变换关系及反常层技巧证明函数方程，完善了该级数的分析性质。
On Landau-Siegel zeros and heights of singular moduli
本文改进Granville-Stark结果，给出L'/L(1,χ_D)与奇异模高度的渐近公式，并基于一致abc猜想推出Landau-Siegel零点不存在的新界，对光滑判别式有改进。
We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As an application, we prove a quadratic analogue of Brun-Titchmarsh theorem on average, bounding the number of primes $p\leqslant X$ with $p^2+1\equiv0\pmod q$. The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by Irving.
Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish explicit formulae for $ψ(x,χ)$, $θ(x,χ)$, and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.
Numerical Computations Concerning Landau-Siegel Zeros
该研究通过新的算法验证了大量模数下二次狄利克雷 $L$-函数在特定区域内的非零性，突破了之前的研究成果。
Primes in Arithmetic Progressions to Large Moduli and Siegel Zeroes
本文在例外特征假设下，将素数在算术级数中的渐近均匀分布推广到模数 $q$ 可达 $x^{2/3-\varepsilon}$，并证明对几乎所有 $a$ 或 $q$ 成立，优于此前 $\theta=30/59$ 或 $16/31$ 的结果。
Dedekind sums and mean square value of $L(1,χ)$ over subgroups
本文研究了在乘法群子群上平凡的奇 Dirichlet 特征对应的 Dirichlet $L$-函数在 $s=1$ 处二次均值的显式公式，探讨了涉及 Dedekind 和的分母相消最优性，并证明了特定素数幂模下 Dedekind 和的无关性。
Dirichlet $L$-functions on the critical line and multiplicative chaos
本文证明随机 Dirichlet 特征对应的 L 函数在临界线上收敛到与高斯乘性混沌相关的随机分布，与 zeta 函数随机平移的极限一致。
Refined Upper Bounds for $L(1,χ)$
本文针对模 $q$ 的非主 Dirichlet 特征，证明了 $L(1,\chi)$ 的一个更精细上界，将误差项改进为 $O(\log\log q/\log q)$，并由此得到关于素数丰富模数的下确界估计。
Traces, high powers and one level density for families of curves over finite fields
本文利用函数域 zeta 函数、显式公式及指定分歧类型素多项式密度，计算高亏格曲线模空间中酉矩阵迹高次幂的期望，并推导多类曲线覆盖的一级密度。
Mertens products in arithmetic progressions over function fields
本文研究了函数域中多项式环F_q[t]上素数算术进展的梅尔滕斯公式，并在无条件情况下得到了类似于GRH强度渐进行为的结果。
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