## The Riemann zeros and the heat equation

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?

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Q. Does anyone have a citation prior to 1994 which mentions this fact?


1988: Numer. Math. 52, 483-497 (the differential equation is given in a slightly different form on page 493).

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In that reference, do you mean the equation: $$H_\lambda(x)=F_\lambda(D)H_0(x),\qquad D=d/dx,$$ where $$F_\lambda(z)=\sum_{m=0}^\infty (-1)^m\lambda^m z^{2m}/m!$$ This is not the heat equation. – Stopple Dec 5 at 0:04
to convert it to the 1994 heat equation, just take the derivative with respect to $\lambda$ of both sides of the 1988 equation, and then note that $\partial F_\lambda/\partial\lambda=-z^2 F_\lambda$. Since $z=d/dx$, you arrive at $\partial H_\lambda/\partial\lambda=-\partial^2 H_\lambda/\partial x^2$, which is the (backward) heat equation (with $\lambda$ representing time). – Carlo Beenakker Dec 5 at 11:39
We could debate whether the derivation above means that the 1988 formula is the heat equation. But regardless I think this answer misses the spirit of the original question, of whether the connection to the heat equation is well known. The word 'heat' does not appear. – Stopple Dec 5 at 16:10