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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 | Abstract: | Let $
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 | Services: | Forum | Review | PDF | Favorites |
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