18 The de Bruijn–Newman constant

A survey on this topic may be found at [ 224 ] .

Let \(H_0 \colon \mathbf{C}\to \mathbf{C}\) denote the function

\begin{equation} \label{hoz} H_0(z) := \frac{1}{8} \xi \left(\frac{1}{2} + \frac{iz}{2}\right), \end{equation}

where \(\xi \) denotes the Riemann xi function

\begin{equation} \label{sas} \xi (s) := \frac{s(s-1)}{2} \pi ^{-s/2} \Gamma \left(\frac{s}{2}\right) \zeta (s) \end{equation}

and \(\zeta \) is the Riemann zeta function. Then \(H_0\) is an entire even function with functional equation \(H_0(\overline{z}) = \overline{H_0(z)}\), and the Riemann hypothesis is equivalent to the assertion that all the zeroes of \(H_0\) are real.

It is a classical fact (see [ 277 , p. 255 ] ) that \(H_0\) has the Fourier representation

\[ H_0(z) = \int _0^\infty \Phi (u) \cos (zu)\ du \]

where \(\Phi \) is the super-exponentially decaying function

\begin{equation} \label{phidef} \Phi (u) := \sum _{n=1}^\infty (2\pi ^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp (-\pi n^2 e^{4u} ). \end{equation}

The sum defining \(\Phi (u)\) converges absolutely for negative \(u\) also. From Poisson summation one can verify that \(\Phi \) satisfies the functional equation \(\Phi (u) = \Phi (-u)\) (i.e., \(\Phi \) is even).

De Bruijn [ 55 ] introduced the more general family of functions \(H_t \colon \mathbf{C}\to \mathbf{C}\) for \(t \in \mathbf{R}\) by the formula

\begin{equation} \label{htdef} H_t(z) := \int _0^\infty e^{tu^2} \Phi (u) \cos (zu)\ du. \end{equation}

As noted in [ 52 , p.114 ] , one can view \(H_t\) as the evolution of \(H_0\) under the backwards heat equation \(\partial _t H_t(z)= -\partial _{zz} H_t(z)\). As with \(H_0\), each of the \(H_t\) are entire even functions with functional equation \(H_t(\overline{z}) = \overline{H_t(z)}\). From results of Pólya [ 239 ] it is known that if \(H_t\) has purely real zeroes for some \(t\) then \(H_{t'}\) has purely real zeroes for all \(t'{\gt}t\). De Bruijn showed that the zeroes of \(H_t\) are purely real for \(t \geq 1/2\). Strengthening these results, Newman [ 223 ] showed that there is an absolute constant \(-\infty {\lt} \Lambda \leq 1/2\), now known as the De Bruijn-Newman constant, with the property that \(H_t\) has purely real zeroes if and only if \(t \geq \Lambda \). The Riemann hypothesis is then clearly equivalent to the upper bound \(\Lambda \leq 0\). Newman conjectured the complementary lower bound \(\Lambda \geq 0\), and noted that this conjecture asserts that if the Riemann hypothesis is true, it is only “barely so”.

Known lower bounds on \(\Lambda \) are listed in the tables below.

Table 18.1 Lower bounds on \(\Lambda \).

Lower bound on \(\Lambda \)	Reference
\({\gt}-\infty \)	Newman 1976 [ 223 ]
\({\gt}-50\)	Csordas–Norfolk–Varga 1988 [ 49 ]
\({\gt}-5\)	te Riele 1991 [ 271 ]
\({\gt}-0.385\)	Norfolk–Ruttan–Varga 1992 [ 226 ]
\({\gt}-0.0991\)	Csordas–Ruttan–Varga 1991 [ 51 ]
\({\gt}-4.379 \times 10^{-6}\)	Csordas–Smith–Varga 1994 [ 52 ]
\({\gt}-5.895 \times 10^{-9}\)	Csordas–Odlyzko–Smith–Varga 1993 [ 50 ]
\({\gt}-2.63 \times 10^{-9}\)	Odlyzko 2000 [ 227 ]
\({\gt}-1.15 \times 10^{-11}\)	Saouter–Gourdon–Demichel 2011 [ 257 ]
\(\geq 0\)	Rodgers–Tao 2020 [ 255 ]
\(\geq 0\)	Dobner 2021 [ 60 ]

The argument of Dobner applies more generally to the Selberg class.

For upper bounds, we have

Table 18.2 Upper bounds on \(\Lambda \).

Upper bound on \(\Lambda \)	Reference
\(\leq 1/2\)	Newman 1976 [ 223 ]
\({\lt} 1/2\)	Ki–Kim–Lee 2009 [ 166 ]
\(\leq 0.22\)	Polymath 2019 [ 241 ]
\(\leq 0.2\)	Platt–Trudgian 2021 [ 238 ]