This is the eleventh 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.

There are currently two strands of activity.  One is writing up the paper describing the combination of theoretical and numerical results needed to obtain the new bound $\Lambda \leq 0.22$.  The latest version of the writeup may be found here, in this directory.  The theoretical side of things have mostly been written up; the main remaining tasks to do right now are

1. giving a more detailed description and illustration of the two major numerical verifications, namely the barrier verification that establishes a zero-free region for $H_t(x+iy)=0$ for $0 \leq t \leq 0.2, 0.2 \leq y \leq 1, |x - 6 \times 10^{10} - 83952| \leq 0.5$, and the Dirichlet series bound that establishes a zero-free region for $t = 0.2, 0.2 \leq y \leq 1, x \geq 6 \times 10^{10} + 83952$; and
2. giving more detail on the conditional results assuming more numerical verification of RH.

Meanwhile, several of us have been exploring the behaviour of the zeroes of $H_t$ for negative $t$; this does not directly lead to any new progress on bounding $\Lambda$ (though there is a good chance that it may simplify the proof of $\Lambda \geq 0$), but there have been some interesting numerical phenomena uncovered, as summarised in this set of slides.  One phenomenon is that for large negative $t$, many of the complex zeroes begin to organise themselves near the curves

$\displaystyle y = -\frac{t}{2} \log \frac{x}{4\pi n(n+1)} - 1.$

(An example of the agreement between the zeroes and these curves may be found here.)  We now have a (heuristic) theoretical explanation for this; we should have an approximation

$\displaystyle H_t(x+iy) \approx B_t(x+iy) \sum_{n=1}^\infty \frac{b_n^t}{n^{s_*}}$

in this region (where $B_t, b_n^t, n^{s_*}$ are defined in equations (11), (15), (17) of the writeup, and the above curves arise from (an approximation of) those locations where two adjacent terms $\frac{b_n^t}{n^{s_*}}$, $\frac{b_{n+1}^t}{(n+1)^{s_*}}$ in this series have equal magnitude (with the other terms being of lower order).

However, we only have a partial explanation at present of the interesting behaviour of the real zeroes at negative t, for instance the surviving zeroes at extremely negative values of $t$ appear to lie on the curve where the quantity $N$ is close to a half-integer, where

$\displaystyle \tilde x := x + \frac{\pi t}{4}$

$\displaystyle N := \sqrt{\frac{\tilde x}{4\pi}}$

The remaining zeroes exhibit a pattern in $(N,u)$ coordinates that is approximately 1-periodic in $N$, where

$\displaystyle u := \frac{4\pi |t|}{\tilde x}.$

A plot of the zeroes in these coordinates (somewhat truncated due to the numerical range) may be found here.

We do not yet have a total explanation of the phenomena seen in this picture.  It appears that we have an approximation

$\displaystyle H_t(x) \approx A_t(x) \sum_{n=1}^\infty \exp( -\frac{|t| \log^2(n/N)}{4(1-\frac{iu}{8\pi})} - \frac{1+i\tilde x}{2} \log(n/N) )$

where $A_t(x)$ is the non-zero multiplier

$\displaystyle A_t(x) := e^{\pi^2 t/64} M_0(\frac{1+i\tilde x}{2}) N^{-\frac{1+i\tilde x}{2}} \sqrt{\frac{\pi}{1-\frac{iu}{8\pi}}}$

and

$\displaystyle M_0(s) := \frac{1}{8}\frac{s(s-1)}{2}\pi^{-s/2} \sqrt{2\pi} \exp( (\frac{s}{2}-\frac{1}{2}) \log \frac{s}{2} - \frac{s}{2} )$

The derivation of this formula may be found in this wiki page.  However our initial attempts to simplify the above approximation further have proven to be somewhat inaccurate numerically (in particular giving an incorrect prediction for the location of zeroes, as seen in this picture).  We are in the process of using numerics to try to resolve the discrepancies (see this page for some code and discussion).