K. Soundararajan
Extending a result of Heath-Brown, we establish an asymptotic formula for the fourth moment of central values of Dirichlet $L$-functions attached to primitive characters $\pmod q$.
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We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We find an asymptotic formula to the fourth moment of the central value of Dirichlet L functions in this context. We also find a lower bound to the 2kth moment of these L-functions.
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.
We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.
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.
We prove an asymptotic formula for the fourth power mean of Dirichlet L-functions averaged over primitive characters to modulus q and over t\in [0,T] which is particularly effective when q \ge T. In this range the correct order of magnitude was not previously known.
We study the fourth moment of quadratic Dirichlet $L$-functions at $s= \frac{1}{2}$. We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally. The proofs of these results follow closely arguments of Soundararajan and Young [19] and Soundararajan [17].
We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.
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
Twisted fourth moment of Dirichlet $L$-functions to a fixed modulus
本文研究固定素数幂模数下 Dirichlet $L$-函数族在临界线上的扭曲四阶矩，并给出带幂次节省误差项的渐近公式。
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.
We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet $L$--functions over $\mathbb{F}_q[x]$, as the base field $\mathbb{F}_q$ is fixed and the genus of the family goes to infinity. According to conjectures of Andrade and Keating, we expect the fourth moment to be asymptotic to $q^{2g+1} P(2g+1)$ up to an error of size $o(q^{2g+1})$, where $P$ is a polynomial of degree $10$ with explicit coefficients. We prove an asymptotic formula with the leading three terms, which agrees with the conjectured result.
The fourth moment of quadratic Dirichlet $L$-Functions II
本研究无条件证明二次Dirichlet $L$-函数四次矩的含四个主项的渐近公式，基于已有工作引入改进大筛估计、交叉抵消分析等新要素，采用精细组合论证完成证明。
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…
We prove results on moments of $L$-functions in the function field setting, where the moment averages are taken over primitive characters of modulus $R$, where $R$ is a polynomial in $\mathbb{F}_q [T]$. We consider the behaviour as $\textrm{deg} R \rightarrow \infty$ and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that $R$ is square-full, and an asymptotic formula for the fourth moment for any $R$. The fourth moment result is a function field analogue of Heath-Brown's result in the number field setting, which was subsequ…
We obtain the asymptotic main term of moments of arbitrary derivatives of $L$-functions in the function field setting. Specifically, the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus $Q \in \mathbb{F}_q [t]$, and the asymptotic limit is as $\mathrm{deg} \, Q \longrightarrow \infty$. This extends the work of Tamam who obtained the asymptotic main term of low moments of $L$-functions, without derivatives, in the function field setting. It also expands on the work of Conrey, Rubinstein, and Snaith who cojectured, using random matr…
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。
We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - ε}$
Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic L-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet L-functions modulo q weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length q^{1/4}. An auxiliary result of independent interest is a sharp upper …
First moment of central values of quadratic Dirichlet $L$-functions
本文利用双重Dirichlet级数方法计算了二次Dirichlet $L$-函数族中心值的一阶矩，并在广义黎曼假设下证明了误差项为主项四次方根的渐近公式。
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 formulate a version of the Random Wave Conjecture for the fourth moment of Eisenstein series which is based on Zagier's regularized inner product. We prove an asymptotic formula expressing the regularized fourth moment as a mean value of L-functions. This is an advantage over previous work in the literature, which has approached the fourth moment problem through truncated Eisenstein series and not yielded a suitable expression in terms of L-functions.
We obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients of holomorphic cusp forms twisted by rational additive twists with small denominators.
We give explicit formulae for all of the terms in the asymptotic expansion of the mean fourth power of the Riemann zeta-function on the critical line.
Conditionally on the Generalized Lindelöf Hypothesis, we obtain an asymptotic for the fourth moment of Hecke Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the Random Wave Conjecture.
We obtain an asymptotic formula for the fourth moment of central values of a family of quadratic Hecke $L$-functions in the Gaussian field under the generalized Riemann hypothesis (GRH). We also establish lower bounds unconditionally and upper bounds under GRH for higher moments of the same family.
The shifted fourth moment of modular form L-functions in the weight aspect
本文研究了模形式L函数在重量方面的移位第四矩，并将其与黎曼ζ函数和自守L函数的矩联系起来，探讨了主项结构以推广到更高矩。
Discrete moments of the Riemann zeta function were studied by Gonek and Hejhal in the 1980's. They independently formulated a conjecture concerning the size of these moments. In 1999, Hughes, Keating, and O'Connell, by employing a random matrix model, made this conjecture more precise. Subject to the Riemann hypothesis, we establish upper and lower bounds of the correct order of magnitude in the case of the fourth moment.
We prove that the fourth moment of holomorphic Hecke cusp forms is bounded provided that the Riemann Hypothesis holds for an appropriate degree 8 L-function. We accomplish this using Watson's formula, which translates the question in hand into a moment problem for L-functions which is amenable to the techniques of Soundararajan and Harper on obtaining sharp bounds for moments of the Riemann zeta function.
We improve the error term in the asymptotic formula for the twisted fourth moment of automorphic L functions of prime level and weight two proved by Kowalski, Michel and Vanderkam. As a consequence, we obtain a new subconvexity bound in the level aspect and improve the lower bound on proportion of simultaneous non-vanishing.
Let $Δ(x)$ and $E(x)$ be error terms of the sum of divisor function and the mean square of the Riemann zeta function, respectively. In this paper their fourth power moments for short intervals of Jutila's type are considered. We get an asymptotic formula for $U$ in some range.
We prove the asymptotic formula for the fourth moment of automorphic $L$-functions of level $p^ν$, where $p$ is a fixed prime number and $ν\rightarrow \infty$. This paper is a continuation of work by Rouymi, who computed asymptotics of the first three moments at prime power level, and a generalization of results obtained for prime level by Duke, Friedlander & Iwaniec and Kowalski, Michel & Vanderkam.
Mean square estimates for $Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, and some related Mellin transforms of quantities connected with the fourth power moment of $|ζ(1/2+ix)|^4$.
We establish unconditional sharp upper bounds of the $k$-th moments of the family of quadratic Dirichlet $L$-functions at the central point for $0 \leq k \leq 2$.
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.
We establish sharp lower bounds for the $k$-th moment in the range $0 \leq k \leq 1$ of the family of quadratic Dirichlet $L$-functions at the central point.
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…
Shifted moments of quadratic Dirichlet $L$-functions
本文在广义黎曼假设下，研究了二次狄利克雷 $L$-函数的移位矩并给出了上界，同时应用该结果推导了二次狄利克雷特征和的矩的界限。
Fourth power moment of twisted Kloosterman sum and Hurwitz class numbers
本文研究扭曲Kloosterman和四次幂矩与Hurwitz类数的关系，借助Calabi-Yau三维簇点计数公式推导显式表达式，利用模形式相关理论得到加权和及该矩的渐近公式。
In this paper, we study moments of the central values of quartic Dirichlet $L$-functions and establish quantitative non-vanishing result for these $L$-values.
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。
We study the third moment of quadratic Dirichlet L-functions, obtaining an error term of size $O(X^{3/4 + \varepsilon})$.
Theta functions, fourth moments of eigenforms, and the sup-norm problem III
本文发展了显式theta测试函数框架以更精确捕捉特征值，将Hecke--Maaß形式四阶矩的尖锐混合界归约为计数问题，并给出了在level aspect上尖锐且对特征值依赖略大于凸界的无条件混合界。
The fourth moment of holomorphic Hecke cusp forms in shorter intervals
本文研究了较短区间内全纯 Hecke 尖形式四次矩的平均值，通过设定区间长度 $H=K^{3/4+c}$，改进了 Khan 的先前结果。
Lower bounds for shifted moments of Dirichlet $L$-functions of fixed modulus
本文在广义黎曼猜想下，建立了固定模Dirichlet $L$-函数双平移矩的精确下界。
We establish lower bounds for the $2k$-th moment of families of quadratic Dirichlet $L$-functions at the central point for all real $k<0$, assuming a conjecture of S. Chowla on the non-vanishing of these $L$-values.
The fourth moment of truncated Eisenstein series
本研究通过引入无代价的截断参数额外平均，得到大拉普拉斯本征值截断艾森斯坦级数四阶矩的渐近式，首次验证其主项符合高斯随机行为，印证了三十多年前Hejhal与Rackner提出的针对艾森斯坦级数的随机波猜想。
An overview of results and problems concerning the asymptotic formula for $\int_0^T|ζ(1/2+it)|^4dt$ is given, together with a discussion of modern methods from spectral theory used in recent work on this subject.
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.
Upper bounds for shifted moments of quadratic Dirichlet $L$-functions over function fields
本文在函数域上建立了二次Dirichlet $L$-函数平移矩的锐利上界，并将其应用于证明二次Dirichlet特征和矩的若干界。
The Laplace transform of $|ζ(1/2+it)|$ is investigated, for which a precise expression is obtained, valid in a certain region in the complex plane. The method of proof is based on complex integration and spectral theory of the non-Euclidean Laplacian.
Upper bounds for shifted moments of Dirichlet $L$-functions to a fixed modulus over function fields
本文建立了函数域上固定模 Dirichlet $L$-函数族平移矩的精确上界，并以此推导出 Dirichlet 特征和矩的上界。
Given $c,$ a positive integer, we give an explicit formula and an asymptotic formula for \[ \sumχ(c)|L(1,\,χ)|^{2}, \] where $χ$ is the non-trivial Dirichlet character mod $f$ with $f>c.$
Amplified Fourth Moment of the Riemann Zeta-Function and Applications
本文针对黎曼ζ函数扭曲四阶矩研究的现有局限，推导得到ζ函数四阶矩乘以放大子四次方的渐近公式，可支撑ζ函数零点间隙、矩下界等相关理论研究。
We study the $2k$-th moment of central values of the family of Dirichlet $L$-functions to a fixed prime modulus. We establish sharp lower bounds for all real $k \geq 0$ and sharp upper bounds for $k$ in the range $0 \leq k \leq 1$.
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。
We study the moments of $\mbox{Tr}(Θ_χ)$ as $χ$ runs over Dirichlet characters defined over $\mathbb{F}_q[T]$ of fixed order $r$. In particular, we show that after an appropriate normalization, the $q$-limit of the power sum moments behave like the power sum moments of the group of unitary matrices multiplied by a weight function.
Mixed fourth moments of automorphic forms and the shifted moments of $L$-functions
本文在GRH和GRC假设下，研究自守形式与Eisentein级数的混合四阶矩，揭示非均匀分布现象并提出相关猜想。
Let $a(n)$ be the Fourier coefficients of a holomorphic cusp form of weight $κ=2n\geqslant12$ for the full modular group and $A(x)=\sum\limits_{n\leqslant x}a(n)$. In this paper, we establish an asymptotic formula of the fourth power moment of $A(x)$ and prove that \begin{equation*} \int_1^TA^4(x)\mathrm{d}x=\frac{3}{64κπ^4}s_{4;2}(\tilde{a}) T^{2κ}+O\big(T^{2κ-δ_4+\varepsilon}\big) \end{equation*} with $δ_4=1/8$, which improves the previous result.
Shifted moments of cubic and quartic Dirichlet $L$-functions
本文在广义黎曼猜想下，建立了三次和四次狄利克雷L函数位移矩的上界，并应用此结果给出了三次和四次狄利克雷特征和的矩界。
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