A lower bound for the de Bruijn-Newman constant \(\Lambda\)
Let \[ H_{\lambda}(x):=\int^{\infty}_{0}e^{\lambda t^ 2}\Phi (t)\cos (xt)\,dt\quad (\lambda \in \mathbb{R}), \] where \[ \Phi (t):=\sum^{\infty}_{n=1}(2n^ 4\pi^ 2e^{9t}-3n^ 2\pi e^{5t})\exp (-n^ 2\pi e^{4t})\quad (0\le t<\infty). \] For any real \(\lambda\), the function \(H_{\lambda}(x)\) is an entire function of order one, which is real fo real x. \textit{N. G. de Bruijn} [Duke Math. J. 17, 197--226 (1950; Zbl 0038.23302)] has shown that (i) \(H_{\lambda}(x)\) has only real zeros for \(\lambda\geq 1/2\), and (ii) if \(H_{\lambda}(x)\) has only real zeros for some real \(\lambda\), then \(H_{\mu}(x)\) also has only real zeros for any \(\mu >\lambda\). \textit{M. Newman} [Proc. Am. Math. Soc. 61, 245--251 (1976; Zbl 0342.42007)] proved that there exists a smallest value of \(\lambda\) for which \(H_{\lambda}(x)\) has only real zeros, in the following sense: there exists a real number \(\Lambda\) such that \(H_{\lambda}(x)\) has only real zeros for \(\lambda\geq \Lambda\), whereas \(H_{\lambda}(x)\) has some nonreal zero for any \(\lambda <\Lambda\). De Bruijn's result i) implies that \(\Lambda\leq 1/2\). The truth of the Riemann hypothesis would imply that \(\Lambda\leq 0\). Newmann (loc. cit.) conjectures that \(\Lambda\geq 0\). In the paper under review \(\Lambda\) is baptized the de Bruijn-Newman constant. (Not only) Because of the relation of \(\Lambda\) with the Riemann hypothesis, it would be interesting to find a lower bound for \(\Lambda\), which is as large as possible. The main purpose of the present paper is to present a constructive method to find such a lower bound. The method is rather involved, and too complicated to be described here. It requires high-precision numerical calculations (with an accuracy of up to 110 decimal digits) on a VAX- 11/780 combined with precise error estimates, to yield the result \(\Lambda >-50\). The authors expect that this bound can be substantially increased at the expense of higher precision calculations and more computer time.
scientific article; zbMATH DE number 4084837
scientific article