View source for De Bruijn–Newman constant

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A constant $\Lambda$ describing the behaviour of functions related to the [[Riemann zeta-function]].  The [[Riemann hypotheses|Riemann hypothesis]] is equivalent to the assertion that $\Lambda \le 0$.  By contrast, Newman conjectured that $\Lambda \ge 0$, and it is known that $\Lambda > −1·14541 \cdot 10^{−11}$.

==Definition==
Let $\Xi$ denote the [[Riemann xi-function]], $\Xi(t) = \xi(\frac12 + it)$ where
$$
\xi(s) = \frac12 s(s-1) \pi^{-s/2} \Gamma(s/2) \zeta(s)
$$
with $\zeta(s)$ the Riemann zeta function.  The Riemann hypothesis is equivalent to the assertion that all the zeroes of $\Xi$ lie on the real line.

We define a family of modified functions $H_\lambda$ by considering $\Xi$ as the [[Fourier transform]] of a function $\Phi(t)$ and defining $H_\lambda$ as the transform of $\Phi(t) \exp(\lambda t^2)$.  The Riemann hypothesis is that $H_0$ has only real zeroes.  De Bruijn proved that $H_\lambda$ has only real zeroes when $\lambda > \frac14$, and Newman showed that while there there exists a $\lambda$ such that $H_\lambda$ has non-real zeroes, there exists a $\Lambda$ such that  $H_\lambda$ has only real zeroes for all $\lambda \ge \Lambda$.  The infimum of such $\Lambda$ is the de Bruijn–Newman constant.

==References==
* Finch, Steven R.  ''Mathematical Constants'', Encyclopedia of Mathematics and Its Applications '''94''', Cambridge University Press (2003) {{ISBN|0-521-81805-2}} {{ZBL|1054.00001}}
* Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick.  "An improved lower bound for the de Bruijn-Newman constant", ''Math. Comput.'' '''80''', No. 276, 2281-2287 (2011) {{DOI|10.1090/S0025-5718-2011-02472-5}} {{ZBL|1267.11094}}