The Riemann xi function $\Xi(x)$ is defined, with $s=1/2+ix$, as
$$
\Xi(x)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)=2\int_0^\infty \Phi(u)\cos(ux) \, du,
$$
where $\Phi(u)$ is defined as
$$
2\sum_{n=1}^\infty\left(2\pi^2n^4\exp(9u/2)-3\pi n^2\exp(5u/2)\right)\exp(-n^2\pi\exp(2u)).
$$
This arises from integration by parts after writing $\Xi$ as the Mellin transform of the theta function, and then a change of variables from multiplicative to additive notation. In 1950 de Bruijn (building on work of Polya) introduced a deformation parameter $t$:
$$
\Xi_t(x)=\int_0^\infty \exp(t u^2)\Phi(u)\cos(ux)\, du,
$$
so that for $t=0$, $\Xi_0(x)$ is just $\Xi(x)/2$.
de Bruijn proved the following theorem about the zeros in $x$:
(i) For $t\ge 1/2$, $\Xi_t(x)$ has only real zeros.
(ii) If for some real $t$, $\Xi_t(x)$ has only real zeros, then $\Xi_{t^\prime}(x)$ also has only real zeros for any $t^\prime>t.$
In 1976 Newman showed that there exists a real constant $\Lambda$, $-\infty<\Lambda\le 1/2$, such that
(i) $\Xi_t(x)$ has only real zeros if and only if $t\ge\Lambda$.
(ii) $\Xi_t(x)$ has some complex zeros if $t<\Lambda$.
The constant $\Lambda$ is known as the de Bruijn-Newman constant. The Riemann hypothesis is the conjecture that $\Lambda\le 0$. Newman made the complementary conjecture that $\Lambda\ge 0$, with the often quoted remark
"This new conjecture is a quantitative
version of the dictum that the Riemann
hypothesis, if true, is only barely
so."
Given the significance of the de Bruijn-Newman constant $\Lambda$, much work has gone into estimating lower bounds, and the current record (Saouter et. al.) is
$
-1.14\times 10^{-11}<\Lambda.
$
A breakthrough occurred in the work of Csordas, Smith and Varga, "Lehmer pairs of zeros, the de Bruijn-Newman constant, and the Riemann Hypothesis", Constructive Approximation, 10 (1994), pp. 107-129.
They realized that unusually close pairs of zeros of the Riemann zeta function, the so-called Lehmer pairs, could be used to give lower bounds on $\Lambda$. The idea of the proof is that the function $\Xi_t(x)$ satisfies the backward heat equation
$$
\frac{\partial \Xi}{\partial t}+\frac{\partial^2 \Xi}{\partial x^2}=0,
$$
from which they are able to draw conclusions about the differential equation satisfied by the $k$-th gap between the zeros as the deformation parameter $t$ varies.
They mention this PDE in a rather offhand way, as a remark on an alternate proof to one of the lemmas. In fact, it does not seem to be well known that the de Bruijn-Newman constant can be interpreted as a time variable in a heat flow. Is this well known? Or put more concretely, does anyone have a citation prior to 1994 which mentions this fact?
Update: Tao and Rodgers have a proof of the Newman conjecture on the arXiv.
Best Answer
1988: Numer. Math. 52, 483-497 (the differential equation is given in a slightly different form on page 493).