Polymath15, fifth thread: finishing off the test problem?

What's new 2018-03-10

This is the fifth “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 almost finished off the test problem of showing that ${H_t(x+iy) \neq 0}$ whenever ${t = y = 0.4}$ . We have two useful approximations for ${H_t}$ , which we have denoted ${A^{eff}+B^{eff}}$ and ${A^{eff}+B^{eff}-C^{eff}}$ , and a normalising quantity ${B^{eff}_0}$ that is asymptotically equal to the above expressions; see the wiki page for definitions. In practice, the ${A^{eff}+B^{eff}}$ approximation seems to be accurate within about one or two significant figures, whilst the ${A^{eff}+B^{eff}-C^{eff}}$ approximation is accurate to about three or four. We have an effective upper bound

$\displaystyle |H_t - A^{eff} - B^{eff}| \leq E_1 + E_2 + E_3^*$

where the expressions ${E_1,E_2,E_3^*}$ are quite small in practice ( ${E_3^*}$ is typically about two orders of magnitude smaller than the main term ${B^{eff}_0}$ once ${x}$ is moderately large, and the error terms ${E_1,E_2}$ are even smaller). See this page for details. In principle we could also obtain an effective upper bound for ${|H_t - (A^{eff} + B^{eff} - C^{eff})|}$ (the ${E_3^*}$ term would be replaced by something smaller).

The ratio ${\frac{A^{eff}+B^{eff}}{B^{eff}_0}}$ takes the form of a difference ${\sum_{n=1}^N \frac{b_n}{n^s} - e^{i\theta} \sum_{n=1}^N \frac{a_n}{n^s}}$ of two Dirichlet series, where ${e^{i\theta}}$ is a phase whose value is explicit but perhaps not terribly important, and the coefficients ${b_n, a_n}$ are explicit and relatively simple ( ${b_n}$ is ${\exp( \frac{t}{4} \log^2 n)}$ , and ${a_n}$ is approximately ${(n/N)^y b_n}$ ). To bound this away from zero, we have found it advantageous to mollify this difference by multiplying by an Euler product ${\prod_{p \leq P} (1 - \frac{b_p}{p^s})}$ to cancel much of the initial oscillation; also one can take advantage of the fact that the ${b_n}$ are real and the ${a_n}$ are (approximately) real. See this page for details. The upshot is that we seem to be getting good lower bounds for the size of this difference of Dirichlet series starting from about ${x \geq 5 \times 10^5}$ or so. The error terms ${E_1,E_2,E_3^*}$ are already quite small by this stage, so we should soon be able to rigorously keep ${H_t}$ from vanishing at this point. We also have a scheme for lower bounding the difference of Dirichlet series below this range, though it is not clear at present how far we can continue this before the error terms ${E_1,E_2,E_3^*}$ become unmanageable. For very small ${x}$ we may have to explore some faster ways to compute the expression ${H_t}$ , which is still difficult to compute directly with high accuracy. One will also need to bound the somewhat unwieldy expressions ${E_1,E_2}$ by something more manageable. For instance, right now these quantities depend on the continuous variable ${x}$ ; it would be preferable to have a quantity that depends only on the parameter ${N = \lfloor \sqrt{ \frac{x}{4\pi} + \frac{t}{16} }\rfloor}$ , as this could be computed numerically for all ${x}$ in the remaining range of interest quite quickly.

As before, any other mathematical discussion related to the project is also welcome here, for instance any summaries of previous discussion that was not covered in this post.