Newman's criterion

Charles Newman [MR 55 #7944], building on work of deBruijn [ MR 12,250] 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 [ MR 2002a:30046] has recently proven that $-2.7·10^{-9}<\Lambda$.

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