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Luis Baez-Duarte

Publications and source records attributed to Luis Baez-Duarte.

12 recordsLinked to original sources
arXiv2005

A divergent Vasyunin correction

V. I. Vasyunin has introduced special sequences of step functions related to the strong Nyman-Beurling criterion that converge pointwise to 1 in $[1,\infty)$. We show here that the first and simplest such sequence considered by Vasyunin diverges in $L_1((1,\infty),x^{-2}dx)$, which of course precludes the $L_2((1,\infty),x^{-2}dx)$-convergence needed for the Riemann hypothesis. Whether all sequences considered by this author also diverge remains an interesting open question.

math.NT↗
arXiv2005

On the spatial mean of the Poincare cycle

Let $X$ be a measure space and $T:X\to X$ a measurable transformation. For any measurable $E\subseteq X$ and $x\in E$, the possibly infinite return time is $n_E(x):=\inf\{n>0: T^n x\in E\}$. If $T$ is an ergodic tranformation of the probability space $X$, and $μ(E)>0$, then a theorem of M. Kac states that $\int_E n_E dμ=1$. We generalize this to any invertible measure preserving transformation $T$ on a finite measure space $X$, by proving independently, and nearly trivially that for any measurable $E\subseteq X$ one has $\int_E n_E dμ=μ(I_E)$, where $I_E$ is the smallest invariant set containing $E$. In particular this also provides a simpler proof of Poincaré's recurrence theorem.

math.PR↗
arXiv2005

A general strong Nyman-Beurling Criterion for the Riemann Hypothesis

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.

math.NT↗
arXiv2005

Moebius-convolutions and the Riemann hypothesis

The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and Hardy-Littlewood, based on the order of growth at infinity along the positive real axis of certain entire functions, are here imbedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion. Some properties of the convolutions involved sharpen what is hitherto known for the Riesz function.

math.NT↗
arXiv2002

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

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 the Riemann hypothesis for some constant $c>0$ the distance between $χ$ and $-\sum_{a=1}^nμ(a)e^{-c\frac{\log a}{\log\log n}}ρ_a$ is of order $(\log\log n)^{-1/3}$.

math.NT↗
arXiv2002

A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis

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 $\{ρ_a | a\in\Nat\}$.

math.NT↗
arXiv2000

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.

math.NT↗