# 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.

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.

## $t\gt0$

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.)

## Bibliography

• [B1950] N. C. de Bruijn, The roots of trigonometric integrals, Duke J. Math. 17 (1950), 197–226.
• [CSV1994] G. Csordas, W. Smith, R. S. Varga, Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis, Constr. Approx. 10 (1994), no. 1, 107–129.
• [KKL2009] H. Ki, Y. O. Kim, and J. Lee, On the de Bruijn-Newman constant, Advances in Mathematics, 22 (2009), 281–306. Citeseer
• [N1976] C. M. Newman, Fourier transforms with only real zeroes, Proc. Amer. Math. Soc. 61 (1976), 246–251.
• [RT2018] B. Rodgers, T. Tao, The de Bruijn-Newman constant is negative, preprint. arXiv:1801.05914