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28 March 2024
 
  » arxiv » math.NT/0205003

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A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis, 2
Luis Baez-Duarte ;
Date 1 May 2002
Subject Number Theory | math.NT
AbstractLet $ ho(x)=x-[x]$, $chi=chi_{(0,1)}$. In $L_2(0,infty)$ consider the subspace $B$ generated by ${ ho_a|ageq1}$ where $ ho_a(x):= ho(frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $chiinar{B}$. 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 $chiinar{Bnat}$ where $Bnat$ is the much smaller subspace generated by ${ ho_a|ainNat}$. This second version differs from the first in showing that under the Riemann hypothesis for some constant $c>0$ the distance between $chi$ and $-sum_{a=1}^nmu(a)e^{-cfrac{log a}{loglog n}} ho_a$ is of order $(loglog n)^{-1/3}$.
Source arXiv, math.NT/0205003
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