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<h1 align=center> Newman's criterion
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Charles Newman [\begin{rawhtml}MR 55 \#7944\end{rawhtml}], 
building on work of deBruijn [
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<a href="http://www.ams.org/msnmain?fn=130&s1=12_250&pg1=MR">MR 12,250</a>
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defined
$$\Xi_\lambda(z)=\int_{-\infty}^\infty \Phi(t)e^{-\lambda t^2}e^{iz} ~dt$$
where 
$$\Phi(t)=2\sum_{n=1}^\infty (2 n^4\pi^2e^{\frac 9 2 t}-3 n^2 \pi e^{\frac 5 2 t})e^{-n^2 \pi e^{2 t}}.$$
Note that $\Xi_0(z)=\Xi(z).$

He proved that there exists a constant $\Lambda$ (with $-1/8 \le \Lambda < \infty$) such that $\Xi_\lambda(z)$ has only real zeros if and only if $\lambda\ge \Lambda$. RH is equivalent to the assertion that $\Lambda\le 0$.

The constant $\Lambda$ (which Newman conjectured is equal to 0) is now called the deBruijn-Newman constant. A. Odlyzko [
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<a href="http://www.ams.org/msnmain?fn=130&s1=2002a:30046&pg1=MR">MR 2002a:30046</a>
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$-2.7·10^{-9}<\Lambda$.
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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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