A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, 2

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…

A note on Nyman's equivalent formulation of the Riemann Hypothesis

A certain subspace of the Hilbert space of square-integrable functions on the unit interval has been considered by Nyman, Beurling, and others, with the result that the constant function 1 belongs to it if and only if the Riemann Hypothesis holds. I show that the product of |1 - 1/rho| taken over the zeros with real parts strictly greater than 1/2, counted with multiplicities, is the norm of the projection of 1 to this subspace. This provides a quantitative refinement to Nyman's theorem.

Arithmetical Aspects of Beurling's Real Variable Reformulation of the Riemann Hypothesis

The paper presents two arithmetical versions of the Nyman-Beurling equivalence with the Riemann hypothesis, proved by classical, quasi elementary, number-theoretic methods, based on an integrated version of the classical combinatorial identity for Moebius numbers. These proofs also give insight into the troublesome phenomenon that many natural sequences of Beurling functions tending to the indicator function of (0,1) both pointwise and in L1 norm do not converge in L2 norm.

Polynomial approximations in a generalized Nyman-Beurling criterion

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis (RH), is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $θ_k\in(0,1)$, $k\ge1$. Randomizing the $θ_k$ generates new structures and criteria. One of them is a sufficient condition for RH that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper generalizes conditions (i) and (ii) that inv…

Beurling–Nyman 几何与 Gram 矩阵结构：通过 Mellin 平滑实现阶梯密度与多项式衰减

Beurling Nyman Geometry and Gram Matrix Structure, Ladder Density and Polynomial Decay via Mellin Smoothing

本文用多尺度阶梯参数化、Mellin 分析与受控平滑算子证明 BN 系统 Gram 非对角元沿阶梯距离具有任意阶多项式衰减，并导出 m>2 时的块压缩性。

Zero free regions for Dirichlet series

In this paper, we are interested in explicit zero-free discs for some Dirichlet series and we also study a general Beurling-Nyman criterion for $L$-functions. Our results generalize and improve previous results obtained by N. Nikolski and by A. de Roton. As a concrete application, we get, for example, a Beurling-Nyman type criterion for the Siegel zero problem.

On an analytic estimate in the theory of the Riemann Zeta function and a Theorem of Baez-Duarte

We establish a uniform upper estimate for the values of zeta(s)/zeta(s+A), 0<= A, on the critical line (conditionally on the Riemann Hypothesis). We use this to give a variant, purely complex analytic, to Baez-Duarte's proof of a strengthened Nyman-Beurling criterion for the validity of the Riemann Hypothesis.

On the maximum of cotangent sums related to the Riemann Hypothesis in rational numbers in short intervals

Cotangent sums play a significant role in the Nyman-Beurling criterion for the Riemann Hypothesis. Here we investigate the maximum of the values of these cotangent sums over various sets of rational numbers in short intervals.

与黎曼猜想等价的插值和外推命题

Interpolation and Extrapolation Statements equivalent to the Riemann Hypothesis

本文通过插值理论提出与黎曼猜想真假等价的命题，并利用Jawerth-Milman外推的关键结果改进Beurling-Nyman准则，给出更精确的充分条件。

Arithmetic and Analysis of the series $\displaystyle { \sum_{n=1}^{\infty} \frac{1}{n} \sin \frac{x}{n} }$

In this paper we connect a celebrated theorem of Nyman and Beurling on the equivalence between the Riemann hypothesis and the density of some functional space in $ L^2(0, 1)$ to a trigonometric series considered first by Hardy and Littlewood. We highlight some of its curious analytical and arithmetical properties.

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…

On General Prime Number Theorems with Remainder

We show that for Beurling generalized numbers the prime number theorem in remainder form $$π(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n\in\mathbb{N}$$ is equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right) \quad \mbox{for all } n \in \mathbb{N},$$ where $N$ and $π$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number th…