This is the eighth “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.

Significant progress has been made since the last update; by implementing the “barrier” method to establish zero free regions for $H_t$ by leveraging the extensive existing numerical verification of the Riemann hypothesis (which establishes zero free regions for $H_0$), we have been able to improve our upper bound on $\Lambda$ from 0.48 to 0.28. Furthermore, there appears to be a bit of further room to improve the bounds further by tweaking the parameters $t_0, y_0, X$ used in the argument (we are currently using $t_0=0.2, y_0 = 0.4, X = 5 \times 10^9$); the most recent idea is to try to use exponential sum estimates to improve the bounds on the derivative of the approximation to $H_t$ that is used in the barrier method, which currently is the most computationally intensive step of the argument.