Neha Elizabeth Thomas, Arya Chandran, K Vishnu Namboothiri
Finding the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $χ$ of same parity is an active problem in number theory. Here we explicitly evaluate such averages of $L(3,χ)$ and $L(4,χ)$ using certain trigonometric sums and Bernoulli polynomials and express them in terms of the Euler totient function $ϕ$ and the Jordan totient function $J_s$.
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继续探索
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.
The Dirichlet Series for the Exterior Square $L$-function on GL(n)
本文给出了GL（n）上的外平方L-函数的Jacquet-Shalika构造的一个初等推导，作为傅里叶系数$A(m_1,...,m_{n-1})$的经典狄利克雷级数.
The mean-square of Dirichlet L-functions
证明了[CFKRS]关于复合模q的Dirichlet L函数临界点附近均方的猜想。当素数模的二阶矩被另一素数上的特征标扭转时，我们还证明了一种倒易公式。
In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet $L$-function with a longer mollifier, whose coefficients are also more general than before. As an application we obtain that, for every Dirichlet $L$-function, more than 41.72\% of zeros are on the critical line and more than 40.74\% of zeros are simple and on the critical line. These proportions also improve previous results which were proved only for the Riemann zeta-function.
Arithmetic Progressions of Squares and Multiple Dirichlet Series
本文研究计数平方数本原三项等差数列的二元多重 Dirichlet 级数，证明其在 $\mathbb{C}^2$ 上的亚纯延拓，并借助 Tauberian 方法获得相关平方数等差数列与圆锥曲线有理点的计数。
The mean square of the product of $ζ(s)$ with Dirichlet polynomials
改进了Balasubramanian，Conrey和Heath-Brown的早期工作，得到了Riemann zeta函数乘以任意长度为$T^{1/2 + δ}$的Dirichlet多项式的均方的渐近公式，并利用$δ= 0.01515....$得到了黎曼ζ函数的三阶矩的正确数量级的上界.我们还完善以前的工作Deshouillers和Iwaniec，获得渐近估计的地方的界限。使用瓦特的工作，我们计算的均方黎曼ζ函数倍的狄利克雷多项式的长度去$T^{3/4}$提供的狄利克雷多项式假定一个特殊的形状。最后，我们给出了Kloosterman分式三线性和的一个几何估计，它包含Lindelof假设。
The mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip
本文改进了Matsumoto和Ishikawa的工作，通过获得Atkinson型误差项显式公式，给出了临界带内黎曼zeta函数与Dirichlet多项式乘积均方值的渐近公式，与广义Dirichlet除数问题紧密相关。
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。
Character Average of Second and Fourth Powers of Dirichlet L-Series at Unity
用简单的初等方法得到了单位点处Dirichlet L-级数四次幂的特征平均值，这一结果似乎是新的。我们还利用单位点处Hurwitz zeta函数第二变量的幂级数得到了单位点处Dirichlet L-级数二次幂的特征平均值。
Exponential sums with Dirichlet coefficients of L-functions
改进和推广了作者最近的结果，给出了区间上所有整数和素数上L函数的Dirichlet系数指数和的条件估计.特别是，我们建立了新的条件结果指数和Hecke特征值和平方的Hecke特征值素数。我们利用这些估计来改进我们最近关于对称平方L函数的Hecke本征形在黎曼假设下的Piatetski-Shapiro素数处的Hecke本征值平方的结果。
Mean square values of Dirichlet $L$-functions associated to fixed order characters
本文研究固定阶特征相关Dirichlet $L$-函数的均方值，建立了其在半临界线上二阶矩的界，给出离散形式并推导了相应的零点密度估计。
Let $π$ be a cuspidal representation on $\GL(2,\mathbb{A}_{\mathbb{Q}}).$ We give nontrivial lower and upper bounds for average of absolute values of Dirichlet coefficients associated to $π;$ and nontrivial upper bound in the case of $\Sym^kπ,$ $k=2, 3.$ These bounds generalize the known estimates in holomorphic case to Maass forms, without assuming Ramanujan-Petersson conjecture.
On the integral of the error term in the Dirichlet divisor problem
得到了几个结果有关的功能$Δ_k(x)$，它代表的误差项在一般狄利克雷除数问题。这些包括该函数的积分估计，以及相应的均方积分。详细研究了$Δ_2(x)$的均方积分。
We consider the asymptotic behavior of the mean square of truncations of the Dirichlet series of $ζ(s)^k$. We discuss the connections of this problem with that of the variance of the divisor function in short intervals and in arithmetic progressions, reviewing the recent results on this topic. Finally, we show how these results can all be proved assuming a suitable version of the moments conjecture.
Siegel's mass formula and averages of Dirichlet L-functions over function fields
设D是F_q[t]中的无平方多项式，其中q是奇数，G是F_q[t]上行列式为D的定三元格的亏格.本文给出了G的类上本原表示数的加权和公式及G的质量公式的完备的、较初等的证明。我们的质量公式的证明显示了一个有趣的关系与某些平均的狄利克雷L函数。
Averages of quadratic twists of long Dirichlet polynomials
本文在Lindelöf假设下计算了由Kronecker符号扭曲的长Dirichlet多项式的平均值，验证了其与[CFKRS]配方中的0-和1-交换项一致。
Averages of long Dirichlet polynomials with modular coefficients
本文在广义 Lindelöf 假设下，研究了原始尖点形式相关 L 函数在权重方面的矩，并得到了带模系数的长 Dirichlet 多项式扭曲矩的渐近公式，该结果与 CFKRS 配方的预测一致。
Factorisation of the product of Dirichlet series of completely multiplicative functions
在第一章中，我们将给出与Dirichlet特征标相关的L函数模的平方值的计算。这个计算建议询问是否存在算术乘法函数的某个环，以及它是否唯一。这种探索导致了第二章中那个环的构建。最后，在第三章中，我们将给出与此环相关的一些命题。下面的结果是本工作的主要结果之一：对于F和G两个完全乘性函数，$ s $一个复数如狄利克雷级数$ D(F,s) $和$ D(G,s) $收敛：$ \forall F,G \in \mathbb{M}_{c} : D(F,s) \times D(G,s) = D(F \times G,2s) \times D(F \square G,s) $其中运算$ \square $在第二章中定义为前面提到的环的和。以下是一些类似的版本，带有$ s = x+iy $：$ \forall F, G \in \mathbb{M}_{c} : ~ D(F,s) \times D(G,\overline{s}) = D(F \times G,2x) \times D(\frac{F}{\text{Id}_{e}^{iy}} \square \frac{G}{\text{Id}_{e}^{-iy}}, x) $ $ \forall F, G \in \mathbb{M}_{c} : ~ |D(F,s)|^{2} = D(|F|^{2},2x) \times D(\frac{F}{\t…
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。
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。
Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet $L$-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square $L$-functions and an asymptotic expansion for the average of central values of generalized Dirichlet $L$-functions.
The eighth moment of Dirichlet $L$-functions
证明了Dirichlet $L$ -函数的八阶矩的一个渐近性，该函数在原始特征$χ$模$q$上平均，在所有模$q\leq Q$上平均，且在临界线上有一个短平均，条件是GRH.我们得到了GL（2）L-函数Dirichlet扭的四阶矩的类似结果。我们的研究结果与文献中的矩量图相匹配;特别是，常数24024出现在八阶矩的首阶项中。
We introduce Möbius functions of higher rank, a new class of arithmetic functions so that the classical Möbius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role.
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。
Approximations of $SL(3,\mathbb{Z})$ Hecke-Maass $L$-Functions by short Dirichlet polynomials
本文通过短Dirichlet多项式近似SL(3,Z) Hecke-Maass L函数，利用Kuznetsov迹公式方法，展示了L函数倒数的平均近似性。
Estimates for character sums and Dirichlet $L$-functions to smooth moduli
我们使用van der Corput方法的$q$ -模拟来估计短特征和以平滑模。若$χ$是模无平方的本原Dirichlet特征标，则$q^δ$ -光滑整数$q$证明了$$L(\frac12,χ)\ll_εq^{\frac{27}{164}+O(δ)+ε}.$$
We prove that $$ \int_1^XΔ(x)Δ_3(x)\,dx \ll X^{13/9}\log^{10/3}X, \quad \int_1^XΔ(x)Δ_4(x)\,dx \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $Δ_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $ζ^k(s)$ ($Δ_2(x) \equiv Δ(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $Δ_k(x)$. We also obtain the analogues of the above bounds when $\D(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|ζ(1/2+it)|$.
On large gaps between consecutive zeros, on the critical line, of some Dirichlet $L$-functions
本文证明了当$χ$是偶本原Dirichlet特征标时，某些Dirichlet $L$ -函数$L(s,χ),$在临界线上存在大的（平均值的3.54倍）连续零点间隙.
Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve
在广义黎曼假设（GRH）下，利用渐近大筛证明了原始Dirichlet $L$ -函数的91%的零点是单零点.这改善了早期的工作厄兹吕克这给了一个比例最多86%。我们进一步计算$q$ -模拟的配对相关函数$F(α)$平均所有原始狄利克雷$L$ -函数的范围$|α| < 2$。以前，只有当平均值包括所有字符$χ$时，这样的结果才可用。
Twisted first moment of central values of primitive quadratic Dirichlet $L$-functions
本文利用双Dirichlet级数与递归论证求值原始二次Dirichlet L-函数中心值的扭曲一阶矩，得到误差项大小为主项平方根的渐近公式。
We study the second moment of Dirichlet $L$-functions to a large prime modulus $q$ twisted by the square of an arbitrary Dirichlet polynomial. We break the $\frac{1}{2}$-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than $q^{51/101} = q^{1/2 +1/202}$. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet $L$-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.
The second moment of Dirichlet twists of Hecke $L$-functions
固定一个Hecke尖点形式$f$，并考虑由一个本原Dirichlet特征标扭曲的$f$的$L$ -函数.当我们在一个大模$q$的所有本原特征标上取值时，这个$L$ -函数的中心值的平方的平均行为是什么？Stefanicki证明了一个渐近有效的只有$q$有很少的素因子，我们将其扩展到几乎所有的$q$。
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。
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.
In this paper, we calculate the absolute tensor square of the Dirichlet $L$-functions and show that it is expressed as an Euler product over pairs of primes. The method is to construct an equation to link primes to a series which has the factors of the absolute tensor product of the Dirichlet $ L $-functions. This study is a generalization of Akatsuka's theorem on the Riemann zeta function, and gives a proof of Kurokawa's prediction proposed in 1992.
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.
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.
The first moment of quadratic Dirichlet L-functions
得到了本原二次Dirichlet L函数在中心点的光滑加权一阶矩的渐近公式，其误差项为主项的平方根.我们的方法使用了一种递归技术，将结果反馈给自己，连续改善误差项。
This paper improves the upper bound for the exceptional zeroes of Dirichlet L-functions with even characters. The result is obtained by improving on explicit estimate for $L'(σ;χ)$ for $σ$ close to unity, using a result on the average of Dirichlet characters, and on the lower bound for $L(1;χ)$, with computational aid.
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。
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。
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。
Average non-vanishing of Dirichlet $L$-functions at the central point
该研究突破广义黎曼假设限制，通过对模q本原特征及区间内q双重平均，无条件证明至少50.073%的Dirichlet L函数中心点值非零，采用三段软化子的软化法，依赖Kloosterman和估计，同时指出存在可修正的错误。
For a positive integer $q\not\equiv 2 \pmod 4$, this work considers the fourth moment of Dirichlet $L$-functions averaged over both $t\in [0,T]$ and primitive characters to modulus $q$. An asymptotic formula with a power saving from both $q$-aspect and $t$-aspect in the error term is obtained.
A note on the fourth moment of Dirichlet L-functions
本文证明了在t \in [0，T]上对模q的本原特征标平均的Dirichlet L函数的四次幂均值的一个渐近公式，当q \ge T时，该公式特别有效.在这个范围内，正确的数量级以前是未知的。
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。
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 $p\nmid q_F$
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.
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 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…
On Divisor Function Polynomials and Dirichlet Series
该研究针对和集系计数函数推导狄利克雷级数与收敛界，定义并研究了除数函数多项式，揭示其与伯努利数、有序贝尔多项式的关联。
Partitions into Sums of Two Squares
本研究通过圆法变体结合鞍点法，解决了两平方和整数分拆数的渐近公式问题，突破了以往受限分拆研究局限，首次处理了带分数奇点的狄利克雷级数
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。
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.
The sixth power moment of Dirichlet L-functions
我们证明了一个公式，与功率节省，狄利克雷L函数的六阶矩平均超过模$q$，超过原始字符$χ$模$q$，并超过临界线。我们的公式完全符合随机矩阵理论的预测。特别地，常数42作为首阶项中的因子出现，完全如对黎曼ζ函数的六阶矩所预测的那样。
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。
On the mean values of Dirichlet L-functions
本文研究了Dirichlet L函数L（s，χ）在临界带中心（s=1/2）的2k次幂矩，其中平均值是所有本原特征标χ（mod q）上的平均值.我们将Gonek、Hughes和Keating关于黎曼ζ函数的混合欧拉-阿达玛积结果扩展到这种情况。这使我们能够恢复基于随机矩阵模型的矩的计算结果，以自然的方式合并算术项。
Numerical estimates on the Landau-Siegel zero and other related quantities
本文对不超过10^7的奇素数模的Dirichlet L-函数进行数值计算，给出L(1,χ)的显式下界和Landau-Siegel零点位置的显式上界，并附带得到虚二次域类数相关界的信息。
Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good error terms gives information on the the location of zeros of L-functions and possible Siegel zeros. Similar results are obtained for an integer in a congruence class expressed as the sum of two primes.
Simultaneous non-vanishing of Dirichlet L-functions
本文在广义黎曼猜想下证明了四个狄利克雷L函数在临界线上同时非消失，且在无条件情况下证明了对于无限多个字符，四个L函数同时非消失，但比例随q趋向无穷而趋于零。
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