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

On Nyman, Beurling and Baez-Duarte's Hilbert space reformulation of the Riemann hypothesis

There has been a surge of interest of late in an old result of Nyman and Beurling giving a Hilbert space formulation of the Riemann hypothesis. Many authors have contributed to this circle of ideas, culminating in a beautiful refinement due to Baez-Duarte. The purpose of this little survey is to dis-entangle the resulting web of complications, and reveal the essential simplicity of the main results.

Zeros of Dirichlet Polynomials via a Density Criterion

We obtain a necessary and sufficient condition in order that a semi-plane of the form $\Re(s)>r$, $r\in \mathbb{R}$, is free of zeros of a given Dirichlet polynomial. The result may be considered a natural generalization of a well-known criterion for the truth of the Riemann hypothesis due to Báez-Duarte. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to Báez-Duarte's one is also established.

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

Riemann Hypothesis: a special case of the Riesz and Hardy-Littlewood wave and a numerical treatment of the Baez-Duarte coefficients up to some billions in the k-variable

本文通过数值研究Riesz和Hardy-Littlewood波（临界函数）在α=15/2、β=4及临界带特定区域内的行为，为黎曼假设在Re(s)>1/2区域的可能成立提供证据。

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 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…

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.

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.

Criteria equivalent to the Riemann Hypothesis

We give a brief overview of a few criteria equivalent to the Riemann Hypothesis. Next we concentrate on the Riesz and B{á}ez-Duarte criteria. We proof that they are equivalent and we provide some computer data to support them. It is not compressed to six pages version of the talk delivered by M.W. during the XXVII Workshop on Geometrical Methods in Physics, 28 June -- 6 July, 2008, Białowie{ż}a, Poland.

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…

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。