For each $f:[0,\infty)\to\Com$ formally consider its co-Poisson or Müntz transform $g(x)=\sum_{n\geq 1}f(nx)-\frac{1}{x}\int_0^\infty f(t)dt$. For certain $f$'s with both $f, g \in L_2(0,\infty)$ it is true that the Riemann hypothesis holds if and only if $f$ is in the $L_2$ closure of the vector space generated by the dilations $g(kx)$, $k\in\Nat$. Such is the case for example when $f=χ_{(0,1]}$ where the above statement reduces to the strong Nyman criterion already established by the author. In this note we show that the necessity implication holds for any continuously differentiable function $f$ vanishing at infinity and satisfying $\int_0^\infty t|f'(t)|dt<\infty$. If in addition $f$ is of compact support then the sufficiency implication also holds true. It would be convenient to remove this compactness condition.
The purpose of the present paper is to provide a general overview of a variety of results related to a category of cotangent sums which have been proven to be associated to the so-called Nyman-Beurling criterion for the Riemann Hypothesis. These sums are also related to the Estermann Zeta function.
We establish generalizations of the Nyman-Beurling and Báez-Duarte criteria concerning lack of zeros of Dirichlet $L$-functions in the semi-plane $\Re(s) >1/p$ for $p\in (1,2]$. We pose and solve a natural extremal problem for Dirichlet polynomials which take values one at the zeros of the corresponding $L$-function on the vertical line $\Re(s)=1/p$.
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.
Let $ρ(x)=x-[x]$, $χ=χ_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{ρ_a | a \geq 1\}$ where $ρ_a(x):=ρ(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $χ\in\bar{\B}$. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that $χ\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{ρ_a | a\in\Nat\}$.
Let $ρ(x)=x-[x]$, $χ=χ_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{ρ_a|a\geq1\}$ where $ρ_a(x):=ρ(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $χ\in\bar{\B}$. For some time it has been conjectured, and proved in the first version of this paper, posted in arXiv:math.NT/0202141 v2, that the Riemann hypothesis is equivalent to the stronger statement that $χ\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{ρ_a|a\in\Nat\}$. This second version differs from the first in showing that under th…
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.
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 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.
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.
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…
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.
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.
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.
In this paper we work out a Riemann-von Mangoldt type formula for the summatory function $ψ(x):=\sum_{g\in G, |g|\le x} Λ_{G}(g)$, where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and $Λ_{G}$ is the corresponding von Mangoldt function attaining $\log|p|$ for $g=p^k$ with a prime element $p\in G$ and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function $ζ_{G}$, belonging to $G$, to the number of zeroes of $ζ_G$ in various regions, in particular within the critical strip where the analytic continuation exists…
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.
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…
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 goal of this paper is to reveal a close connection between the following three subjects that have not been studied together in the past: terminal and canonical cyclic quotient singularities, integer ratios of factorials, Nyman's approach to the Riemann Hypothesis. In particular, we notice that the constructions of P.A. Picon are relevant for the study of singularities and possibly the Riemann Hypothesis. The list of the 29 stable quintuples of Mori-Morrison-Morrison coincides, up to the choice of notation, with the list of the 29 step functions with five terms of Vasyunin. We also reformul…
It is a well-known fact that Riemann Hypothesis will follows if the function identically equal to -1 can be arbitrarily approximated in the norm $\norma{.}$ of $L^{2}([0,1],dx)$ by functions of the form $f(x)=\sum_{k=1}^{n}a_{k} ρ(\frac{θ_{k}}{x})$, where $ρ(x)\adef\pfrac{x}$, and $a_{k}\in\cc$, $0<θ_{k}\leqslant 1$ satisfies $\sum_{k=1}^{n}a_{k} θ_{k}=0$. Parsevall Identity $\norma{f(x)+1}^{2}=\sum_{n\in\zz}\modulo{c(n)}^{2}$ is a possible tool to compute or estimate this norm. In this note we give an expression for the Fourier coefficients $c(n)$ of $f+1$, when $f$ is a function defined as a…
Let $πS(t)$ denote the argument of the Riemann zeta-function, $ζ(s)$, at the point $s=\frac{1}{2}+it$. Assuming the Riemann hypothesis, we present two proofs of the bound $$ |S(t)| \leq \left(\tfrac{1}{4} + o(1) \right)\tfrac{\log t}{\log \log t} $$ for large $t$. This improves a result of Goldston and Gonek by a factor of 2. The first method consists in bounding the auxiliary function $S_1(t) = \int_0^{t} S(u) du$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log\log t…
