16 The de Bruijn–Newman constant

A survey on this topic may be found at [ 175 ] .

Let $H_{0} : C \to C$ denote the function

H_{0} (z) := \frac{1}{8} ξ (\frac{1}{2} + \frac{i z}{2}),

where $ξ$ denotes the Riemann xi function

ξ (s) := \frac{s (s - 1)}{2} π^{- s / 2} Γ (\frac{s}{2}) ζ (s)

and $ζ$ is the Riemann zeta function. Then $H_{0}$ is an entire even function with functional equation $H_{0} (\overset{―}{z}) = \overset{―}{H_{0} (z)}$ , and the Riemann hypothesis is equivalent to the assertion that all the zeroes of $H_{0}$ are real.

It is a classical fact (see [ 217 , p. 255 ] ) that $H_{0}$ has the Fourier representation

H_{0} (z) = \int_{0}^{\infty} Φ (u) \cos (z u) d u

where $Φ$ is the super-exponentially decaying function

Φ (u) := \sum_{n = 1}^{\infty} (2 π^{2} n^{4} e^{9 u} - 3 π n^{2} e^{5 u}) \exp (- π n^{2} e^{4 u}) .

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

De Bruijn [ 45 ] introduced the more general family of functions $H_{t} : C \to C$ for $t \in R$ by the formula

H_{t} (z) := \int_{0}^{\infty} e^{t u^{2}} Φ (u) \cos (z u) d u .

As noted in [ 42 , p.114 ] , one can view $H_{t}$ as the evolution of $H_{0}$ under the backwards heat equation $\partial_{t} H_{t} (z) = - \partial_{z z} H_{t} (z)$ . As with $H_{0}$ , each of the $H_{t}$ are entire even functions with functional equation $H_{t} (\overset{―}{z}) = \overset{―}{H_{t} (z)}$ . From results of Pólya [ 187 ] it is known that if $H_{t}$ has purely real zeroes for some $t$ then $H_{t^{'}}$ has purely real zeroes for all $t^{'} > t$ . De Bruijn showed that the zeroes of $H_{t}$ are purely real for $t \geq 1 / 2$ . Strengthening these results, Newman [ 174 ] showed that there is an absolute constant $- \infty < Λ \leq 1 / 2$ , now known as the De Bruijn-Newman constant, with the property that $H_{t}$ has purely real zeroes if and only if $t \geq Λ$ . The Riemann hypothesis is then clearly equivalent to the upper bound $Λ \leq 0$ . Newman conjectured the complementary lower bound $Λ \geq 0$ , and noted that this conjecture asserts that if the Riemann hypothesis is true, it is only “barely so”.

Known lower bounds on $Λ$ are listed in the tables below.

Table 16.1 Lower bounds on

Λ

Lower bound on $Λ$	Reference
$> - \infty$	Newman 1976 [ 174 ]
$> - 50$	Csordas–Norfolk–Varga 1988 [ 39 ]
$> - 5$	te Riele 1991 [ 212 ]
$> - 0.385$	Norfolk–Ruttan–Varga 1992 [ 176 ]
$> - 0.0991$	Csordas–Ruttan–Varga 1991 [ 41 ]
$> - 4.379 \times 10^{- 6}$	Csordas–Smith–Varga 1994 [ 42 ]
$> - 5.895 \times 10^{- 9}$	Csordas–Odlyzko–Smith–Varga 1993 [ 40 ]
$> - 2.63 \times 10^{- 9}$	Odlyzko 2000 [ 177 ]
$> - 1.15 \times 10^{- 11}$	Saouter–Gourdon–Demichel 2011 [ 201 ]
$\geq 0$	Rodgers–Tao 2020 [ 199 ]
$\geq 0$	Dobner 2021 [ 47 ]

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

For upper bounds, we have

Table 16.2 Upper bounds on

Λ

Upper bound on $Λ$	Reference
$\leq 1 / 2$	Newman 1976 [ 174 ]
$< 1 / 2$	Ki–Kim–Lee 2009 [ 131 ]
$\leq 0.22$	Polymath 2019 [ 188 ]
$\leq 0.2$	Platt–Trudgian 2021 [ 186 ]