The fourth moment of the zeta-function

A New Approach to the Spectral Theory of the Fourth Moment of the Riemann Zeta-Function

The aim of the present work is to exhibit a new proof of the explicit spectral expansion for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in the sense that it dispenses completely with the Kloostermania, the spectral theory of sums of Kloosterman sums that was used in the former proof. The argument is now constructed precisely upon the spectral structure of the Lie group PSL(2,R). Main ingredients in our argument are the theory of automorphic representations as well as the harmonic analysis on the bi…

The moments of the Riemann zeta-function. Part I: The fourth moment off the critical line

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 $|ζ(σ+it)|$.

Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field

We prove the Kloosterman-Spectral sum formula for PSL(2,Z[i])\PSL(2,C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. This sum formula allows the extension of the spectral theory of Kloosterman sums to all algebraic number fields.

Moments of zeta and correlations of divisor-sums: I

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.

关于黎曼zeta函数四次矩误差项的研究

The error term of the fourth moment of the Riemann zeta-function

本文改进了黎曼zeta函数四次矩误差项及其平方积分的对数幂次界，从而对8至12次矩获得小的对数改进，尤其对12次幂矩有适度提升。

Moments of zeta and correlations of divisor-sums: II

This is part II of our examination of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations.

Critical zeros of the Riemann zeta-function

In this unpublished note, we sketch an idea of using a three-piece mollifier to slightly improve the known percentages of zeros and simple zeros of the Riemann zeta-function on the critical line. This uses the recent result of Bettin, Bui, Li and Radziwill [BBLR] on the twisted fourth moment of the Riemann zeta-function.

Mean Values of Zeta-Functions via Representation Theory

The aim of the present article is to reveal a structure shared by two basic zeta-functions in their fourth power moments through the view point of representation theory of Lie groups, relying specifically upon the Kirillov model. It might induce one to ponder over the possibility to go beyond.

A quadratic divisor problem and moments of the Riemann zeta-function

We estimate asymptotically the fourth moment of the Riemann zeta-function twisted by a Dirichlet polynomial of length $T^{\frac14 - \varepsilon}$. Our work relies crucially on Watt's theorem on averages of Kloosterman fractions. In the context of the twisted fourth moment, Watt's result is an optimal replacement for Selberg's eigenvalue conjecture. Our work extends the previous result of Hughes and Young, where Dirichlet polynomials of length $T^{\frac{1}{11}-\varepsilon}$ were considered. Our result has several applications, among others to the proportion of critical zeros of the Riemann zeta…

On the integral of the fourth Jacobi theta function

We generalize the Raabe-formula to the $q$-loggamma function. As a consequence, we get that the integral of the logarithm of the fourth Jacobi theta function between its least imaginary zeros is connected to the partition function and the Riemann zeta function.

Weighted fourth moments of Hecke zeta functions with groessencharacters

We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |ζ(1/2+it,λ^d)|^4| \sum_{0<|μ|^2\leq M} A(μ)λ^d((μ)) |μ|^{-2it}|^2 {\rm d}t$, where $λ^d$ is the groessencharacter satisfying $λ^d((α)) = λ^d(α{\Bbb Z}[i]) = (α/|α|)^{4d}$, for $0\neqα\in{\Bbb Z}[i]$, and $ζ(s,λ^d)$ is the Hecke zeta function that satisfies $ζ(s,λ^d) =(1/4)\sum_{0\neqα\in{\Bbb Z}[i]} λ^d((α)) |α|^{-2s}$ for $\Re(s)>1$, while the numbers $D,M\in(0,\infty)$ and function $A:{\Bbb Z}[i]-\{0\}\rightarrow{\Bbb C}$ are arbitrary (though it is only in respect of cases in wh…

On the $a$-points of symmetric sum of multiple zeta function

In this paper, we present some results on the $a$-points of the symmetric sum of the Euler-Zagier multiple zeta function. Our first three results are for the $a$-points free region of the function. The fourth result is the Riemann-von Mangoldt type formula. In the last two results, we study the real parts of $a$-points of the function.

On the fourth power moment of $Δ(x)$ and $E(x)$ in short intervals

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.

On some conjectures and results for the Riemann zeta-function and Hecke series

Some results and conjectures on $Z_2(s) = \int_1^\infty |ζ(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are presented. Consequences of these conjectures regarding the eighth moment of $|ζ(1/2+it)$ and the error term in the fourth moment of $|ζ(1/2+it)$ are discussed.

The fourth moment of ζ^{'}(ρ)

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.

Sharp bound for the fourth moment of holomorphic Hecke cusp forms

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.

On the splitting conjecture in the hybrid model for the Riemann zeta function

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.

zeta和的矩的上界

Upper bounds for moments of zeta sums

该研究在黎曼假设前提下，借助黎曼zeta函数平移矩的成果，推导出zeta和的矩的上界，为数论领域相关研究提供了新的理论支撑。

模形式L函数在重量方面的移位第四矩

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。

The fourth moment of individual Dirichlet L-functions on the critical line

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.

Exponential moments of the logarithm of the Riemann zeta-function twisted by arguments

We discuss moments of the Riemann zeta-function in this paper. The purpose of this paper is to give an upper bound of exponential moments of the logarithm of the Riemann zeta-function twisted by arguments. Our results contain an improvement of Najnudel result for exponential moments of the argument of the Riemann zeta-function and an unconditional upper bound of the moments.

The Twisted Second Moment of the Dedekind Zeta Function of a Quadratic Field

We compute the second moment of the Dedekind zeta function of a quadratic field times an arbitrary Dirichlet polynomial of length $T^{1/11-ε}$.

黎曼zeta函数矩的上界中隐式常数的优化

Optimization of the Implicit Constant for Upper Bounds for Moments of the Riemann zeta Function

本文优化了Harper证明的黎曼zeta函数矩的精细上界中的隐式常数，并在特定条件下计算了Soundararajan证明的上界中的隐式常数。

A conjecture for the sixth power moment of the Riemann zeta-function

The authors conjecture an asymptotic expression for the sixth power moment of the Riemann zeta function. They establish related results on the asymptotics of the zeta function that support the conjecture.

The eighth moment of the Riemann zeta function

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.

Lower bounds for moments of the derivative of the Riemann zeta function

We establish in this paper sharp lower bounds for the $2k$-th moment of the derivative of the Riemann zeta function on the critical line for all real $k \geq 0$.

Mixed moments of the Riemann zeta function

We analyse a collection of mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$ conjecture.

Discrete Moments of the Riemann Zeta-Function near $σ=1$

We employ mean value estimates of Weyl sums in order to obtain discrete second moments of the Riemann Zeta-function with respect to polynomials near the vertical line $1+i\mathbb{R}$.

Riemann zeta函数在其局部极值处的矩

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。

Higher moments of distribution of zeta zeros

We develop a method for mean-value estimation of long Dirichlet polynomials. For an application, we use our method to study properties of the logarithmic derivative of the Riemann zeta function.

Moments of the Riemann zeta-function

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in families of $L$-functions.

Random matrix theory and discrete moments of the Riemann zeta function

We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta function, and provide a uniform approach to understanding moments of the zeta function and its derivative.

黎曼zeta函数的平移矩

Shifted moments of the Riemann zeta function

本文证明黎曼假设蕴含Chandee关于黎曼zeta函数平移矩的猜想，证明基于Harper关于临界线上$2k$次矩精确上界的思想。

Sharp upper bounds for fractional moments of the Riemann zeta function

We establish sharp upper bounds for the $2k$th moment of the Riemann zeta function on the critical line, for all real $0 \leqslant k \leqslant 2$. This improves on earlier work of Ramachandra, Heath-Brown and Bettin-Chandee-Radziwiłł

Some explicit and unconditional results on gaps between zeroes of the Riemann zeta-function

We make explicit an argument of Heath-Brown concerning large and small gaps between nontrivial zeroes of the Riemann zeta-function, $ζ(s)$. In particular, we provide the first unconditional results on gaps (large and small) which hold for a positive proportion of zeroes. To do this we prove explicit bounds on the second and fourth power moments of $S(t+h)-S(t)$, where $S(t)$ denotes the argument of $ζ(s)$ on the critical line and $h \ll 1 / \log T$. We also use these moments to prove explicit results on the density of the nontrivial zeroes of $ζ(s)$ of a given multiplicity.

Lower bounds for moments of zeta prime rho

Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $ζ(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan that provides analogous bounds for moments of $L$-functions at the central point, averaged over families.

ζ和的高阶矩的下界

Lower bounds for high moments of zeta sums

本文研究和式∑_{n≤x}n^{it}的高阶矩，证明了该高阶矩的无条件下界。

Lower bounds for discrete negative moments of the Riemann zeta function

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.

Lower bounds for moments of zeta and $L$-functions revisited

This paper describes a method to compute lower bounds for moments of $ζ$ and $L$-functions. The method is illustrated in the case of moments of $|ζ(\frac 12+it)|$, where the results are new for small moments $0< k<1$.

黎曼zeta函数导数的复矩

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。

Riemann Zeta 函数扭曲二阶矩的一个互反关系

A reciprocity relation for the twisted second moment of the Riemman Zeta function

本文证明了 Riemann Zeta 函数扭曲二阶矩的互反关系，为 Dirichlet L 函数的 Conrey 公式提供了类比。

黎曼ζ函数导数的负离散矩

Negative discrete moments of the derivative of the Riemann zeta-function

该研究在数论领域，推导黎曼ζ函数导数负离散矩在满密度零点子族上的条件上界，k≤1/2时2k阶矩界近乎最优，还在相关猜想下得到更大零点子族上同等强度的负矩上界。

Computing the moment polynomials of the zeta function

We describe a method to accelerate the numerical computation of the coefficients of the polynomials $P_k(x)$ that appear in the conjectured asymptotics of the $2k$-th moment of the Riemann zeta function. We carried out our method to compute the moment polynomials for $k \leq 13$, and used these to experimentally test conjectures for the moments up to height $10^8$.

Moments of the Riemann zeta-function at its relative extrema on the critical line

Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The proof requires upper and lower bounds for continuous moments of derivatives of Riemann zeta-function on the critical line.

Upper bounds for the moments of zeta prime rho

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $ζ(s)$ for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments. The proof is based upon a recent method of K. Soundararajan that provides analogous bounds for continuous moments of the Riemann zeta-function as well as for moments L-functions at the central point, averaged over families.

On the second moment of $S(T)$ in the theory of the Riemann zeta function

We assume the Riemann Hypothesis and an quantitative form of the Twin Prime Conjecture, and obtain an asymptotic formula for the second moment of $S(T)$ with better error term.

Riemann zeta函数的放大矩

Amplified moments of the Riemann zeta function

建立了Riemann zeta函数的两片放大第二和第四矩的渐近公式，并应用于无条件获得zeta的若干联合矩的有效下界，与Keating-Wei和Keating-Snaith的猜想高度一致。

Upper bounds for fractional joint moments of the Riemann zeta function

We establish upper bounds for the joint moments of the $2k^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative for $0 \leq h \leq 1$ and $1 \leq k \leq 2$. These bounds are expected to be sharp based upon predictions from random matrix theory.

An upper bound for discrete moments of the derivative of the Riemann zeta-function

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.

Twisted mixed moments of the Riemann zeta function

We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape $P(\log T) T^{C}$ for a suitable constant $C<1$ and a polynomial $P(x)$. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$-conjecture.

Moments of zeta and correlations of divisor-sums: stratification and Vandermonde integrals

We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

The second shifted moment of the Riemann zeta function

We prove an asymptotic formula for the second moment (up to height $T$) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to $T^{2-ε}$, for any $ε>0.$

High moments of the Riemann zeta-function

The authors describe a general approach which, in principal, should produce the correct (conjectural) formula for every even integer moment of the Riemann zeta function. They carry it out for the sixth and eigth powers; in the case of sixth powers this leads to the formula conjectured by Conrey and Ghosh, and in the case of eighth powers is new.

The sixth moment of the Riemann zeta function and ternary additive divisor sums

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…