# De Bruijn-Newman constant

For each real number $t$, define the entire function $H_t: {\mathbf C} \to {\mathbf C}$ by the formula

$\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du$

where $\Phi$ is the super-exponentially decaying function

$\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(-\pi n^2 e^{4u}).$

It is known that $\Phi$ is even, and that $H_t$ is even, real on the real axis, and obeys the functional equation $H_t(\overline{z}) = \overline{H_t(z)}$. In particular, the zeroes of $H_t$ are symmetric about both the real and imaginary axes. One can also express $H_t$ in a number of different forms, such as

$\displaystyle H_t(z) = \frac{1}{2} \int_{\bf R} e^{tu^2} \Phi(u) e^{izu}\ du$

or

$\displaystyle H_t(z) = \frac{1}{2} \int_0^\infty e^{t\log^2 x} \Phi(\log x) e^{iz \log x}\ \frac{dx}{x}.$

In the notation of [KKL2009], one has

$\displaystyle H_t(z) = \frac{1}{2} \Xi_{4t}(2z).$
• Note: there may be a typo in the definition of $\Xi_\lambda$ in [KKL2009], they may instead have intended to write $4\lambda (\log x)^2 + 2 it \log x$ in place of $\frac{\lambda}{4} (\log x)^2 + \frac{it}{2} \log x$ in that definition. If so, the relationship would be $H_t(z) = \frac{1}{2} \Xi_{t/4}(z/2)$ instead of $H_t(z) = \frac{1}{2} \Xi_{4t}(2z)$.

De Bruijn [B1950] and Newman [N1976] showed that there existed a constant, the de Bruijn-Newman constant $\Lambda$, such that $H_t$ has all zeroes real precisely when $t \geq \Lambda$. The Riemann hypothesis is equivalent to the claim that $\Lambda \leq 0$. Currently it is known that $0 \leq \Lambda \lt 1/2$ (lower bound in [RT2018], upper bound in [KKL2009]).

## $t=0$

When $t=0$, one has

$\displaystyle H_0(z) = \frac{1}{8} \xi( \frac{1}{2} + \frac{iz}{2} )$

where

$\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} \Gamma(s/2) \zeta(s)$

is the Riemann xi function. In particular, $z$ is a zero of $H_0$ if and only if $\frac{1}{2} + \frac{iz}{2}$ is a non-trivial zero of the Riemann zeta function. Thus, for instance, the Riemann hypothesis is equivalent to all the zeroes of $H_0$ being real, and Riemann-von Mangoldt formula (in the explicit form given by Backlund) gives

$\displaystyle N_0(T) - (\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} - \frac{7}{8})| \lt 0.137 \log (T/2) + 0.443 \log\log(T/2) + 4.350$

for any $T \gt 4$, where $N_0(T)$ denotes the number of zeroes of $H_0$ with real part between 0 and T.

The first $10^{13}$ zeroes of $H_0$ (to the right of the origin) are real [G2004]. This numerical computation uses the Odlyzko-Schonhage algorithm.

## $t\gt0$

For any $t\gt0$, it is known that all but finitely many of the zeroes of $H_t$ are real and simple [KKL2009, Theorem 1.3]

Let $\sigma_{max}(t)$ denote the largest imaginary part of a zero of $H_t$, thus $\sigma_{max}(t)=0$ if and only if $t \geq \Lambda$. It is known that the quantity $\frac{1}{2} \sigma_{max}(t)^2 + t$ is non-decreasing in time whenever $\sigma_{max}(t)\gt0$ (see [KKL2009, Proposition A]. In particular we have

$\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2$

for any $t$.

The zeroes $z_j(t)$ of $H_t$ (formally, at least) obey the system of ODE

$\partial_t z_j(t) = - \sum_{k \neq j} \frac{2}{z_k(t) - z_j(t)}$

where the sum may have to be interpreted in a principal value sense. (See for instance [CSV1994, Lemma 2.4]. This lemma assumes that $t \gt \Lambda$, but it is likely that one can extend to other $t \geq 0$ as well.)

In [KKL2009, Theorem 1.4], it is shown that for any fixed $t\gt0$, the number $N_t(T)$ of zeroes of $H_t$ with real part between 0 and T obeys the asymptotic

$N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + t \log T + O(1)$

as $T \to \infty$ (caution: the error term here is not uniform in t). Also, the zeroes behave like an arithmetic progression in the sense that

$z_{k+1}(t) - z_k(t) = (1+o(1)) \frac{4\pi}{\log |z_k|(t)} = (1+o(1)) \frac{4\pi}{\log k}$

as $k \to +\infty$.
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