D. R. Heath-Brown
Let $k$ be a positive real number, and let $M_k(q)$ be the sum of $|L(\tfrac12,χ)|^{2k}$ over all non-principal characters to a given modulus $q$. We prove that $M_k(q)\ll_k ϕ(q)(\log q)^{k^2}$ whenever $k$ is the reciprocal $n^{-1}$ of a positive integer $n$. If one assumes the Generalized Riemann Hypothesis then the estimate holds for all positive real $k<2$.
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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 $L(1/2, χ)$.
A well-known result, due to Dirichlet and later generalized by de la Vallee-Poussin, expresses a relationship between the sum of fractional parts and the Euler-Mascheroni constant. In this paper, we prove an asymptotic relationship between the summation of the products of fractional parts with powers of integers on one hand, and the values of the Riemann zeta function, on the other hand. Dirichlet's classical result falls as a particular case of this more general theorem.
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.
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 $\pmod q$.
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.
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。
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.
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Bounds for moments of quadratic Dirichlet character sums of prime moduli
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We study the third moment of quadratic Dirichlet L-functions, obtaining an error term of size $O(X^{3/4 + \varepsilon})$.
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.
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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。
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 $f>c.$
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 $0 \leq k \leq 1$.
We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.
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.
Twisted fourth moment of Dirichlet $L$-functions to a fixed modulus
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First moment of central values of quadratic Dirichlet $L$-functions
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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。
Lower bounds for negative moments of Dirichlet $L$-functions to a fixed modulus
本文在固定素数模的本原Dirichlet L函数族中心值非零的假设下，针对所有实k<0，建立了其2k次负矩的下界。
The mixed second moment of prime quadratic Dirichlet $L$-functions over function fields
该研究在函数域背景下，针对首一不可约多项式对应的二次Dirichlet L-函数的μ阶和ν阶导数，计算出其混合二阶矩的渐近公式，为相关L-函数导数矩的研究提供了结果。
Bounds for moments of quadratic Dirichlet character sums
本文研究平滑二次 Dirichlet 特征和的矩估计问题，在广义 Riemann 假设下证明其上界，从而确认 M. Jutila 的相关猜想。
The Vladimirov-Taibleson Operator: Inequalities, Dirichlet Problem, Boundary Hölder Regularity
本文研究了非阿基米德局部域上的Vladimirov-Taibleson伪微分算子，证明了分数阶拉普拉斯算子经典不等式的类似结果，并探讨了Dirichlet问题及解的边界Hölder正则性。
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