The Polymath15 paper “Effective approximation of heat flow evolution of the Riemann ${\xi}$ function, and a new upper bound for the de Bruijn-Newman constant“, submitted to Research in the Mathematical Sciences, has just been uploaded to the arXiv. This paper records the mix of theoretical and computational work needed to improve the upper bound on the de Bruijn-Newman constant ${\Lambda}$. This constant can be defined as follows. The function $\displaystyle H_0(z) := \frac{1}{8} \xi\left(\frac{1}{2} + \frac{iz}{2}\right),$

where ${\xi}$ is the Riemann ${\xi}$ function $\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{-s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$

has a Fourier representation $\displaystyle H_0(z) = \int_0^\infty \Phi(u) \cos(zu)\ du$

where ${\Phi}$ is the super-exponentially decaying function $\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp(-\pi n^2 e^{4u} ).$

The Riemann hypothesis is equivalent to the claim that all the zeroes of ${H_0}$ are real. De Bruijn introduced (in different notation) the deformations $\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du$

of ${H_0}$; one can view this as the solution to the backwards heat equation ${\partial_t H_t = -\partial_{zz} H_t}$ starting at ${H_0}$. From the work of de Bruijn and of Newman, it is known that there exists a real number ${\Lambda}$ – the de Bruijn-Newman constant – such that ${H_t}$ has all zeroes real for ${t \geq \Lambda}$ and has at least one non-real zero for ${t < \Lambda}$. In particular, the Riemann hypothesis is equivalent to the assertion ${\Lambda \leq 0}$. Prior to this paper, the best known bounds for this constant were $\displaystyle 0 \leq \Lambda < 1/2$

with the lower bound due to Rodgers and myself, and the upper bound due to Ki, Kim, and Lee. One of the main results of the paper is to improve the upper bound to $\displaystyle \Lambda \leq 0.22. \ \ \ \ \ (1)$

At a purely numerical level this gets “closer” to proving the Riemann hypothesis, but the methods of proof take as input a finite numerical verification of the Riemann hypothesis up to some given height ${T}$ (in our paper we take ${T \sim 3 \times 10^{10}}$) and converts this (and some other numerical verification) to an upper bound on ${\Lambda}$ that is of order ${O(1/\log T)}$. As discussed in the final section of the paper, further improvement of the numerical verification of RH would thus lead to modest improvements in the upper bound on ${\Lambda}$, although it does not seem likely that our methods could for instance improve the bound to below ${0.1}$ without an infeasible amount of computation.

We now discuss the methods of proof. An existing result of de Bruijn shows that if all the zeroes of ${H_{t_0}(z)}$ lie in the strip ${\{ x+iy: |y| \leq y_0\}}$, then ${\Lambda \leq t_0 + \frac{1}{2} y_0^2}$; we will verify this hypothesis with ${t_0=y_0=0.2}$, thus giving (1). Using the symmetries and the known zero-free regions, it suffices to show that $\displaystyle H_{0.2}(x+iy) \neq 0 \ \ \ \ \ (2)$

whenever ${x \geq 0}$ and ${0.2 \leq y \leq 1}$.

For large ${x}$ (specifically, ${x \geq 6 \times 10^{10}}$), we use effective numerical approximation to ${H_t(x+iy)}$ to establish (2), as discussed in a bit more detail below. For smaller values of ${x}$, the existing numerical verification of the Riemann hypothesis (we use the results of Platt) shows that $\displaystyle H_0(x+iy) \neq 0$

for ${0 \leq x \leq 6 \times 10^{10}}$ and ${0.2 \leq y \leq 1}$. The problem though is that this result only controls ${H_t}$ at time ${t=0}$ rather than the desired time ${t = 0.2}$. To bridge the gap we need to erect a “barrier” that, roughly speaking, verifies that $\displaystyle H_t(x+iy) \neq 0 \ \ \ \ \ (3)$

for ${0 \leq t \leq 0.2}$, ${x = 6 \times 10^{10} + O(1)}$, and ${0.2 \leq y \leq 1}$; with a little bit of work this barrier shows that zeroes cannot sneak in from the right of the barrier to the left in order to produce counterexamples to (2) for small ${x}$.

To enforce this barrier, and to verify (2) for large ${x}$, we need to approximate ${H_t(x+iy)}$ for positive ${t}$. Our starting point is the Riemann-Siegel formula, which roughly speaking is of the shape $\displaystyle H_0(x+iy) \approx B_0(x+iy) ( \sum_{n=1}^N \frac{1}{n^{\frac{1+y-ix}{2}}} + \gamma_0(x+iy) \sum_{n=1}^N \frac{n^y}{n^{\frac{1+y+ix}{2}}} )$

where ${N := \sqrt{x/4\pi}}$, ${B_0(x+iy)}$ is an explicit “gamma factor” that decays exponentially in ${x}$, and ${\gamma_0(x+iy)}$ is a ratio of gamma functions that is roughly of size ${(x/4\pi)^{-y/2}}$. Deforming this by the heat flow gives rise to an approximation roughly of the form $\displaystyle H_t(x+iy) \approx B_t(x+iy) ( \sum_{n=1}^N \frac{b_n^t}{n^{s_*}} + \gamma_t(x+iy) \sum_{n=1}^N \frac{n^y}{n^{\overline{s_*}}} ) \ \ \ \ \ (4)$

where ${B_t(x+iy)}$ and ${\gamma_t(x+iy)}$ are variants of ${B_0(x+iy)}$ and ${\gamma_0(x+iy)}$, ${b_n^t := \exp( \frac{t}{4} \log^2 n )}$, and ${s_*}$ is an exponent which is roughly ${\frac{1+y-ix}{2} + \frac{t}{4} \log \frac{x}{4\pi}}$. In particular, for positive values of ${t}$, ${s_*}$ increases (logarithmically) as ${x}$ increases, and the two sums in the Riemann-Siegel formula become increasingly convergent (even in the face of the slowly increasing coefficients ${b_n^t}$). For very large values of ${x}$ (in the range ${x \geq \exp(C/t)}$ for a large absolute constant ${C}$), the ${n=1}$ terms of both sums dominate, and ${H_t(x+iy)}$ begins to behave in a sinusoidal fashion, with the zeroes “freezing” into an approximate arithmetic progression on the real line much like the zeroes of the sine or cosine functions (we give some asymptotic theorems that formalise this “freezing” effect). This lets one verify (2) for extremely large values of ${x}$ (e.g., ${x \geq 10^{12}}$). For slightly less large values of ${x}$, we first multiply the Riemann-Siegel formula by an “Euler product mollifier” to reduce some of the oscillation in the sum and make the series converge better; we also use a technical variant of the triangle inequality to improve the bounds slightly. These are sufficient to establish (2) for moderately large ${x}$ (say ${x \geq 6 \times 10^{10}}$) with only a modest amount of computational effort (a few seconds after all the optimisations; on my own laptop with very crude code I was able to verify all the computations in a matter of minutes).

The most difficult computational task is the verification of the barrier (3), particularly when ${t}$ is close to zero where the series in (4) converge quite slowly. We first use an Euler product heuristic approximation to ${H_t(x+iy)}$ to decide where to place the barrier in order to make our numerical approximation to ${H_t(x+iy)}$ as large in magnitude as possible (so that we can afford to work with a sparser set of mesh points for the numerical verification). In order to efficiently evaluate the sums in (4) for many different values of ${x+iy}$, we perform a Taylor expansion of the coefficients to factor the sums as combinations of other sums that do not actually depend on ${x}$ and ${y}$ and so can be re-used for multiple choices of ${x+iy}$ after a one-time computation. At the scales we work in, this computation is still quite feasible (a handful of minutes after software and hardware optimisations); if one assumes larger numerical verifications of RH and lowers ${t_0}$ and ${y_0}$ to optimise the value of ${\Lambda}$ accordingly, one could get down to an upper bound of ${\Lambda \leq 0.1}$ assuming an enormous numerical verification of RH (up to height about ${4 \times 10^{21}}$) and a very large distributed computing project to perform the other numerical verifications.

This post can serve as the (presumably final) thread for the Polymath15 project (continuing this post), to handle any remaining discussion topics for that project.