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<h1 align=center> The Beurling-Nyman Criterion
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In his 1950 thesis [
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<a href="http://www.ams.org/msnmain?fn=130&s1=12_108g&pg1=MR">MR 12,108g</a>
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$\mathcal N_{(0,1)} $ is dense in $L^2(0,1).$
Here, $\mathcal N_{(0,1)} $ is the space of functions 
$$f(t)=\sum_{k=1}^n c_k \rho(\theta_k/t)$$
for which $\theta_k\in (0,1)$ and such that $\sum_{k=1}^n c_k=0$.

Beurling [
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<a href="http://www.ams.org/msnmain?fn=130&s1=17_15a&pg1=MR">MR 17,15a</a>
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(1) $\zeta(s)$ has nozeros in $\sigma>1/p$

(2)   $\mathcal N_{(0,1)} $ is dense in  $L^p(0,1)$
(3) The characteristic function $\chi_{(0,1)}$ is in the closure 
of $\mathcal N_{(0,1)} $   in  $L^p(0,1)$
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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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