This is the seventh “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant ${\Lambda}$, continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

The most recent news is that we appear to have completed the verification that ${H_t(x+iy)}$ is free of zeroes when ${t=0.4}$ and ${y \geq 0.4}$, which implies that ${\Lambda \leq 0.48}$. For very large ${x}$ (for instance when the quantity ${N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor}$ is at least ${300}$) this can be done analytically; for medium values of ${x}$ (say when ${N}$ is between ${11}$ and ${300}$) this can be done by numerically evaluating a fast approximation ${A^{eff} + B^{eff}}$ to ${H_t}$ and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of ${x}$, in part due to some new, and significantly faster, numerical ways to evaluate ${H_t}$ in this range.

One obvious thing to do now is to experiment with lowering the parameters ${t}$ and ${y}$ and see what happens. However there are two other potential ways to bound ${\Lambda}$ which may also be numerically feasible. One approach is based on trying to exclude zeroes of ${H_t(x+iy)=0}$ in a region of the form ${0 \leq t \leq t_0}$, ${X \leq x \leq X+1}$ and ${y \geq y_0}$ for some moderately large ${X}$ (this acts as a “barrier” to prevent zeroes from flowing into the region ${\{ 0 \leq x \leq X, y \geq y_0 \}}$ at time ${t_0}$, assuming that they were not already there at time ${0}$). This require significantly less numerical verification in the ${x}$ aspect, but more numerical verification in the ${t}$ aspect, so it is not yet clear whether this is a net win.

Another, rather different approach, is to study the evolution of statistics such as ${S(t) = \sum_{H_t(x+iy)=0: x,y>0} y e^{-x/X}}$ over time. One has fairly good control on such quantities at time zero, and their time derivative looks somewhat manageable, so one may be able to still have good control on this quantity at later times ${t_0>0}$. However for this approach to work, one needs an effective version of the Riemann-von Mangoldt formula for ${H_t}$, which at present is only available asymptotically (or at time ${t=0}$). This approach may be able to avoid almost all numerical computation, except for numerical verification of the Riemann hypothesis, for which we can appeal to existing literature.

Participants are also welcome to add any further summaries of the situation in the comments below.