Related papers: On probabilistic generalizations of the Nyman-Beur…
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…
The Riemann hypothesis (RH) is a long-standing open problem in mathematics. It conjectures that non-trivial zeros of the zeta function all have real part equal to 1/2. The extent of the consequences of RH is far-reaching and touches a wide…
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…
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.…
We give a conditional result on the constant in the B\'aez-Duarte reformulation of the Nyman-Beurling criterion for the Riemann Hypothesis. We show that assuming the Riemann hypothesis and that $\sum_{\rho}\frac{1}{|\zeta'(\rho)|^2}\ll…
We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis. These sums contain the M\"obius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably…
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…
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…
According to two remarkable theorems of Nyman and B\'aez-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…
Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a | a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the…
A crucial role in the Nyman-Beurling-B\'aez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty\left|1-\zeta…
For each $f:[0,\infty)\to\Com$ formally consider its co-Poisson or M\"{u}ntz 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…
Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a…
This papers presents a generalization of the Weitzman overlapping coefficient, originally defined for two probability density functions, to a setting involving k independent distributions, denoted by Delta. To estimate this generalized…
Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a|a\geq1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the…
We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the…
We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function.…
One of the approaches to the Riemann Hypothesis is the Nyman-Beurling criterion. Cotangent sums play a significant role. Here we investigate the values of these cotangent sums for various shifts of the argument.
Under the Riemann Hypothesis, we connect the distribution of $k$-free numbers with the derivative of the Riemann zeta-function at nontrivial zeros of $\zeta(s)$. Moreover, with additional assumptions, we prove the existence of a limiting…
Dilation is a puzzling phenomenon within Imprecise Probability theory: when it obtains, our uncertainty evaluation on event $A$ is vaguer after conditioning $A$ on $B$, whatever is event $B$ in a given partition $\mathcal{B}$. In this paper…