This is the ninth “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.

We have now tentatively improved the upper bound of the de Bruijn-Newman constant to ${\Lambda \leq 0.22}$. Among the technical improvements in our approach, we now are able to use Taylor expansions to efficiently compute the approximation ${A+B}$ to ${H_t(x+iy)}$ for many values of ${x,y}$ in a given region, thus speeding up the computations in the barrier considerably. Also, by using the heuristic that ${H_t(x+iy)}$ behaves somewhat like the partial Euler product ${\prod_p (1 - \frac{1}{p^{\frac{1+y-ix}{2}}})^{-1}}$, we were able to find a good location to place the barrier in which ${H_t(x+iy)}$ is larger than average, hence easier to keep away from zero.

The main remaining bottleneck is that of computing the Euler mollifier bounds that keep ${A+B}$ bounded away from zero for larger values of ${x}$ beyond the barrier. In going below ${0.22}$ we are beginning to need quite complicated mollifiers with somewhat poor tail behavior; we may be reaching the point where none of our bounds will succeed in keeping ${A+B}$ bounded away from zero, so we may be close to the natural limits of our methods.

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