De Bruijn–Newman constant

2020 Mathematics Subject Classification: Primary: 11M [MSN][ZBL]

A constant $\Lambda$ describing the behaviour of functions related to the Riemann zeta-function. The 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