Nathan Ng
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.
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We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function averaged over the zeros of the zeta-function that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.
Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta
本文推广了Landau-Gonek定理，研究了zeta函数零点处的和，并通过近似函数方程计算离散矩，进而证明了Shanks猜想。
We show that the splitting conjecture in the hybrid model of Gonek--Hughes--Keating holds to order on the Riemann hypothesis. Our results are valid in a larger range of the parameter $X$ which mediates between the partial Euler and Hadamard products. We also show that the asymptotic splitting conjecture holds for this larger range of $X$ in the cases of the second and fourth moments.
Let the summatory function of the Möbius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. The main results shown are that the weak Mertens conjecture and the existence of a limiting distribution of $e^{-y/2}M(e^{y})$ are consequences of the aforementioned conjectures. By probabilistic techniques, we present an argument that suggests $M(x)$ grows as large positive and large negative as a constant times $\pm \sqrt{x} (\log \log \log…
In this article, we establish an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture. This builds on the work of the first author on the sixth moment of the Riemann zeta function and work of Conrey-Gonek and Ivić. A key input is a sharp bound for a certain shifted moment of the Riemann zeta function, assuming the Riemann hypothesis.
We prove lower bounds for the discrete negative $2k$th moment of the derivative of the Riemann zeta function for all fractional $k\geqslant 0$. The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general formula for the discrete twisted second moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith.
We develop a hybrid Euler-Hadamard product model for quadratic Dirichlet $L$--functions over function fields (following the model introduced by Gonek, Hughes and Keating for the Riemann-zeta function). After computing the first three twisted moments in this family of $L$--functions, we provide further evidence for the conjectural asymptotic formulas for the moments of the family.
Assuming the Riemann hypothesis, we establish an upper bound for the $2k$-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where $k$ is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating, and O'Connell. This sharpens a result of Milinovich. Our proof builds upon a method of Adam Harper concerning continuous moments of the zeta-function on the critical line.
The discrete second moment of mixed derivatives of the Riemann zeta function
本文研究了黎曼ζ函数在ζ零点处混合导数的离散二次矩的渐进行为，扩展了Conrey--Snaith和Milinovich的工作，提供了无条件和有条件误差项。
Complex moments of the derivative of the Riemann zeta function
本文通过两种随机矩阵方法（Selberg积分精确公式与混合方法）猜想黎曼zeta函数导数在非平凡零点处的复矩结果。
We study the 2k-th power moment of Dirichlet L-functions L(s,χ) at the centre of the critical strip (s=1/2), where the average is over all primitive characters χ(mod q). We extend to this case the hybrid Euler-Hadamard product results of Gonek, Hughes & Keating for the Riemann zeta function. This allows us to recover conjectures for the moments based on random matrix models, incorporating the arithmetical terms in a natural way.
Moments of the Riemann zeta function at its local extrema
本文结合前人研究，计算了Riemann zeta函数及其导数在临界线上局部极值处的第一矩，给出完整渐近展开，并考虑了函数方程中的因子。
On the joint second moment of zeta and its logarithmic derivative
本文在假设Riemann Hypothesis下，研究了zeta函数及其对数导数的加权联合二阶矩，为距临界线$a/\log T$处的该二阶矩提供了上下界。
Assuming the Riemann Hypothesis we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $ζ(s)$. For example, integrating $|ζ(1/2+α+it)|^{-2k}$ with respect to $t$ from $T$ to $2T$, we obtain an asymptotic formula when the shift $α$ is roughly bigger than $\frac{1}{\log T}$ and $k < 1/2$. We also obtain non-trivial upper bounds for much smaller shifts, as long as $\log\frac{1}α \ll \log \log T$. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture …
Keating and Snaith modeled the Riemann zeta-function ζ(s) by characteristic polynomials of random NxN unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of ζ(1/2+it) when k>-1/2. However, an arithmetical factor, widely believed to be part of the leading term coefficient, had to be inserted in an ad hoc manner. Gonek, Hughes and Keating later developed a hybrid formula for ζ(s) that combines a truncation of its Euler product with a product over its zeros. Using it, they recovered the moment conjecture of Keating and Snaith in a way that naturally include…
Moments of derivatives of the Riemann zeta function: Characteristic polynomials and the hybrid formula
本文通过圆酉群特征多项式导数矩的渐近式与混合模型两种途径，提出黎曼zeta函数混合导数在非平凡零点处取值的矩猜想。
An effective Linear Independence conjecture for the zeros of the Riemann zeta function and applications
本文基于对数线性型理论中的启发式论证，提出黎曼zeta函数零点纵坐标的有效线性无关性猜想，并在此猜想及Gonek-Hejhal猜想下，得到素数定理误差项和Möbius函数求和函数的Omega结果，条件性解决Gonek部分猜想。
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 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.
We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - ε}$
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.
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.
We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term.
Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. Since then no other moments have been asymptotically evaluated. In this article we study the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula with power savings error term for the sixth moment of th…
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。
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 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.
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 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].
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 …
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.
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.
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。
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.
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。
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 $|ζ(1/2+ix)|^4$.
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。
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。
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.
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.
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.
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。
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。
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…
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.
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…
In this paper, the first part of a larger work, we prove the spectral decomposition of $$ \int_{-\infty}^\infty|ζ(\s+it)|^4g(t){\rm d}t\qquad(\hf < σ< 1 {\rm {fixed}}), $$ where $g(t)$ is a suitable weight function of fast decay. This is used to obtain estimates and omega results for the function $$\eqalign{E_2(T,σ) &: =\int_0^T|ζ(σ+it)|^4{rm d}t - {ζ^4(2σ)\overζ(4σ)}T -{T\over3-4σ}{({T\over2π} )}^{2-4σ}{ζ^4(2-2σ)\overζ(4-4σ)}\cr& - T^{2-2σ}(a_0(σ) + a_1(σ)\log T + a_2(σ)\log^2T),\cr} $$ the error term in the asymptotic formula for the fourth moment of $|ζ(σ+it)|$.
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。
Spectral fourth moments of Hecke--Maaß cusp forms
研究\(\mathrm{GL}(2)\)自守形式谱矩，考虑谱参数在二进区间的赫克 - 马阿斯尖形式族于黑格纳点的第四矩，建立最优界并应用于移位卷积问题及逐点丢番图指数。
If $$ Δ(x) := \sum_{n\le x}c_n - Cx $$ denotes the error term in the classical Rankin-Selberg problem, then it is proved that $$ \int_0^X Δ^4(x)\d x \ll_εX^{3+ε},\quad \int_0^X Δ_1^4(x)\d x \ll_εX^{11/2+ε}, $$ where $Δ_1(x) = \int_0^xΔ(u) du$. The latter bound is, up to `$ε$', best possible.
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…
The fourth moment of the Hurwitz zeta function
该研究针对平移参数为无理性指数小于3的无理数的Hurwitz zeta函数，给出其临界线上四阶矩的精确上界，确定了0≤k≤2时2k阶矩的量级，推测其临界线上值分布为高斯分布。
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