New Versions of the Nyman-beurling Criterion for the Riemann Hypothesis

@inproceedings{BezDuarte2002NewVO,
  title={New Versions of the Nyman-beurling Criterion for the Riemann Hypothesis},
  author={Luis B{\'a}ez-Duarte},
  year={2002}
}
Let ρ(x) = x− [x], χ = χ(0,1), λ(x) = χ(x) logx, and M(x) = ∑ k≤x μ(k), where μ is the Möbius function. Norms are in Lp(0,∞), 1 < p < ∞. For M1(θ) = M(1/θ) it is noted that ζ(s) ≠ 0 in s > 1/p is equivalent to ‖M1‖r < ∞ for all r ∈ (1,p). The space is the linear space generated by the functions x ρ(θ/x) with θ ∈ (0,1]. Define Gn(x)= ∫ 1 1/nM1(θ)ρ(θ/x)θ−1dθ. For all p ∈ (1,∞) we prove the two arithmetical versions of the Nyman-Beurling theorem: (I) ‖M1‖p <∞ implies λ∈ Lp , and λ∈ Lp implies ‖M1… CONTINUE READING

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