黎曼假设：Riesz和Hardy-Littlewood波的一个特例及对Baez-Duarte系数在k变量中数十亿量级的数值处理

An arithmetical function related to Báez-Duarte's criterion for the Riemann hypothesis

In this mainly expository article, we revisit some formal aspects of B{á}ez-Duarte's criterion for the Riemann hypothesis. In particular, starting from Weingartner's formulation of the criterion, we define an arithmetical function $ν$, which is equal to the M{ö}bius function if, and only if the Riemann hypothesis is true. We record the basic properties of the Dirichlet series of $ν$, and state a few questions. KEYWORDS: Riemann hypothesis, arithmetical functions, Dirichlet series, Hilbert space

On the Riesz and Baez-Duarte criteria for the Riemann Hypothesis

We investigate the relation between the Riesz and the B{á}ez-Duarte criterion for the Riemann Hypothesis. In particular we present the relation between the function $R(x)$ appearing in the Riesz criterion and the sequence $c_k$ appearing in the B{á}ez-Duarte formulation. It is shown that $R(x)$ can be expressed by $c_k$, and, vice versa, the sequence $c_k$ can be obtained from the values of $R(x)$ at integer arguments. Also, we give some relations involving $c_k$ and $R(x)$, and value of the alternating sum of $c_k$.

Báez-Duarte's Criterion for the Riemann Hypothesis and Rice's Integrals

Criterion for the Riemann hypothesis found by Báez-Duarte involves certain real coefficients $c_{k\text{}}$defined as alternating binomial sums. These coefficients can be effectively investigated using Nö% rlund-Rice's integrals. Their behavior exhibits characteristic trend, due to trivial zeros of zeta, and fading oscillations, due to complex zeros. This method enables to calculate numerical values of $c_{k\text{}}$for large values of $k$, at least to $k=4\cdot 10^{8}$. We give explicit expressions both for the trend and for the oscillations. The first tends to zero and is therefore, in view …

关于贝兹 - 杜阿尔特准则的几点注记

A few remarks on the Baez-Duarte Criterion

该论文围绕贝兹 - 杜阿尔特准则展开，核心方法是推导相关引理，主要贡献是得出有趣引理。

Une étude asymptotique probabiliste des coefficients d'une série entière

Following the ideas of Rosenbloom [7] and Hayman [5], Luis Báez-Duarte gives in [1] a probabilistic proof of Hardy-Ramanujan's asymptotic formula for the partitions of an integer. The main principle of the method relies on the convergence in law of a family of random variables to a gaussian variable. In our work we prove a theorem of the Liapounov type (Chung [2]) that justifies this convergence. To obtain simple asymptotic formulæ a condition of the so-called strong Gaussian type defined by Luis Báez-Duarte is required; we demonstrate this in a situation that make it possible to obtain a clas…

A lower bound in an approximation problem involving the zeros of the Riemann zeta function

We slightly improve the lower bound of Baez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the so-called `Hilbert-Polya idea'.

On probabilistic generalizations of the Nyman-Beurling criterion for the zeta function

The Nyman-Beurling criterion is an approximation problem in the space of square integrable functions on $(0,\infty)$, which is equivalent to the Riemann hypothesis. This involves dilations of the fractional part function by factors $θ_k\in(0,1)$, $k\ge1$. We develop probabilistic extensions of the Nyman-Beurling criterion by considering these $θ_k$ as random: this yields new structures and criteria, one of them having a significant overlap with the general strong Báez-Duarte criterion. We start here the study of these criteria, with a special focus on exponential and gamma distributions. The m…

The criteria of Riesz, Hardy-Littlewood et al. for the Riemann Hypothesis revisited using similar functions

The original criteria of Riesz and of Hardy-Littlewood concerning the truth of the Riemann Hypothesis (RH) are revisited and further investigated in light of the recent formulations and results of Maslanka and of Baez-Duarte concerning a representation of the Riemann Zeta function. Then we introduce a general set of similar functions with the emergence of Poisson-like distributions and we present some numerical experiments which indicate that the RH may barely be true.

椭圆模 $j$-函数 Fourier 系数渐近公式的另一证明

An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular $j$-function

本文基于概率方法，利用具有简单无穷乘积的模函数表示，证明了椭圆模 $j$-函数系数的渐近公式。

An optimal choice of Dirichlet polynomials for the Nyman-Beurling criterion

We give a conditional result on the constant in the Báez-Duarte reformulation of the Nyman-Beurling criterion for the Riemann Hypothesis. We show that assuming the Riemann hypothesis and that $\sum_ρ\frac{1}{|ζ'(ρ)|^2}\ll T^{3/2-δ}$, for some $δ>0$, the value of this constant coincides with the lower bound given by Burnol.

A positivity conjecture related to the Riemann zeta function

According to two remarkable theorems of Nyman and Báez-Duarte, the Riemann hypothesis is equivalent to a simply-stated criterion concerning least-squares approximation. In carrying out computations related to this criterion, we have observed a curious phenomenon: for no apparent reason, at least the first billion entries of a certain infinite triangular matrix associated to the Riemann zeta function are all positive. In this article we describe the background leading to this observation, and make a conjecture.

On a Hilbert Space Reformulation of Riemann Hypothesis

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, Báez-Duarte, et. al. with a weighted Bergman space $\mathcal{H}=A_1^2(\mathbb{D})$, i.e., Riemann hypothesis holds if and only if the Hilbert subspace $\mathcal{H}_0$ spanned by a certain family of functions coincides with $\mathcal{H}$. A condition that a function does not belong to $\mathcal{H}_0^\bot$ is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid $T_\mathbb{N}=\{T_k:\, k\in \mathbb{N}\}$ of operators is exactly $B(\mathcal{H})$. As a result, Riemann hypot…

Riemann zeta 函数的无零点区域与加权 Dirichlet 空间中的逼近

Zero-free regions of the Riemann zeta function and approximation in weighted Dirichlet spaces

本文研究黎曼Zeta函数在加权Dirichlet空间中的无零点区域，证明了对标准加权空间D_α(α∈(-3,-2))的逼近问题可给出半平面无零点条件，并将结果推广至解析函数空间ℓ^p_α，特例情形给出了素数定理的新等价表述。