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

2002/05/01 by Luis Báez‐Duarte, Luis Baez-Duarte, Baez-Duarte, Luis · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #Spectral Theory in Mathematical Physics #math.NT

paper · pdf · doi:10.48550/arxiv.math/0205003

9 pages

arxiv created 2002/05/01 · openalex publication_date 2002/05/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ρ(x)=x-[x], χ=χ_(0,1). In L₂(0,∞) consider the subspace \B generated by \ρ_a|a≥1\ where ρ_a(x):=ρ((1)/(ax)). By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement χ∈\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∈\Bnat where \Bnat is the much smaller subspace generated by \ρ_a|a∈\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 -∑_a=1ⁿμ(a)e^{-c(log a)/(loglog n)}ρ_a is of order (loglog n)^-1/3.

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