Approximating Newman's de Bruijn constant

Goal

Approximate the de Bruijn–Newman constant Λ by studying the heat flow of the Riemann zeta function.

Overview

1. Define these equations, describing the heat flow of the Riemann Equation
   1. $H_t(z):={\int }_0^{\infty }{e}^{t{u}^2}\varphi (u)\cos (zu)du$
   2. $\Phi (u):={\sum }_{n=1}^{\infty }(2{\pi }^2{n}^4{e}^{9u}-3\pi n^2{e}^{5u})\exp (-\pi n^2{e}^{4u})$
2. $\exists \Lambda \in \mathbb{R}$ such that $\forall t\ge \Lambda$, the zeroes of $H_t$ are real. This is known as the de Bruijn–Newman constant.
3. $\Lambda \ge 0$ was proved by Brad Rogers & Terence Tau.
4. $\Lambda >0$ is equivalent to the Riemann Hypothesis.
5. $\Lambda \le .22$ was proved as part of the polymath project.
6. $\Lambda \le .2$ was later proved by Dave Platt and Tim Trudgian
7. $H_0(z)={\int }_0^{\infty }\varphi (u)cos(zu)du$ is the Fourier representation of $H_0(z)$
8. $H_0$ relates to the zeroes of Riemann’s zeta function
9. $H_t$ is the evolution of $H_0$ under the backwards heat equation ${\partial }_t{H}_t(z)=-{\partial }_{zz}{H}_t(z)$.

Properties of the Riemann Zeta

1. $\zeta (s)={\sum }_{n=1}^{\infty }\frac{1}{n^s}={\prod }_p[1-p^{-s}{]}^{-1}$
2. $\zeta$ satisfies the functional equation ${\pi }^{-\frac{s}{2}}\Gamma (\frac{s}{2})\zeta (s)={\pi }^{-\frac{1-s}{2}}\Gamma (\frac{1-s}{2})\zeta (1-s))$

Plan to bound $\Lambda$

1. Choose a $t\ge \Lambda$ ( $t>.2$ suffices for the time being )
2. Take $n$ zeroes $(x_1,x_2,\dots ,x_n)$ of $H_t$
3. Computationally estimate how the $x_i$ evolve backwards in time (i.e. towards $t=0$, getting an estimate of the time $t_k$ at which the trajectories of the various $x_i$ converge to a single point
4. Reproduce the above with $n\to \infty$, obtaining better estimates of $\Lambda$.
5. If
   1. $G\approx 0$, we proved nothing 😢.
   2. $G$ is statistically far from $0$, we have shown that Riemann’s Hypothesis is likely true 🎉

Pre-requisites

Finding zeros

What are the best methods known to find zeroes of the $H_t$. I have no clue how to find zeroes given the lack of knowledge about the properties of $\Phi$ and $\zeta$ , namely bounds on the change rate of the function.

• Plot the function $H_t(z)$ for $t=.22$ and $z\in \mathbb{R}$. Visualise these zeroes. I can do this visually, but I have no confidence this gives me a good approximation of zeroes due to continuity concerns.

Modelling the time evolution of the zeroes

Do we have examples ( in code or books ) of approximating path trajectories using heatflow?Andrea suggested Runge-Kutta. Sounds good

Next steps

Both of the points below follow Andrea’s email suggestion

• Study how to use the Runge-Kutta methodology to solve differential equations. This will be needed to propagate the solution’s of zeroes at $H_t$ backwards in time and find when do the zeroes collapse ( i.e. approximating the $\Lambda$ defined below )
• Find numerical methods to approximate the zeroes of $H_t$ , for a given large $t$, i.e. $\ge .22$.