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This is the third “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this previous thread. 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 are making progress on the following test problem: can one show that whenever , , and ? This would imply that

which would be the first quantitative improvement over the de Bruijn bound of (or the Ki-Kim-Lee refinement of ). Of course we can try to lower the two parameters of later on in the project, but this seems as good a place to start as any. One could also potentially try to use finer analysis of dynamics of zeroes to improve the bound further, but this seems to be a less urgent task.

Probably the hardest case is , as there is a good chance that one can then recover the case by a suitable use of the argument principle. Here we appear to have a workable Riemann-Siegel type formula that gives a tractable approximation for . To describe this formula, first note that in the case we have

and the Riemann-Siegel formula gives

for any natural numbers , where is a contour from to that winds once anticlockwise around the zeroes of but does not wind around any other zeroes. A good choice of to use here is

In this case, a classical steepest descent computation (see wiki) yields the approximation

where

Thus we have

where

with and given by (1).

Heuristically, we have derived (see wiki) the more general approximation

for (and in particular for ), where

In practice it seems that the term is negligible once the real part of is moderately large, so one also has the approximation

For large , and for fixed , e.g. , the sums converge fairly quickly (in fact the situation seems to be significantly better here than the much more intensively studied case), and we expect the first term

of the series to dominate. Indeed, analytically we know that (or ) as (holding fixed), and it should also be provable that as well. Numerically with , it seems in fact that (or ) stay within a distance of about of once is moderately large (e.g. ). This raises the hope that one can solve the toy problem of showing for by numerically controlling for small (e.g. ), numerically controlling and analytically bounding the error for medium (e.g. ), and analytically bounding both and for large (e.g. ). (These numbers and are arbitrarily chosen here and may end up being optimised to something else as the computations become clearer.)

Thus, we now have four largely independent tasks (for suitable ranges of “small”, “medium”, and “large” ):

- Numerically computing for small (with enough accuracy to verify that there are no zeroes)
- Numerically computing for medium (with enough accuracy to keep it away from zero)
- Analytically bounding for large (with enough accuracy to keep it away from zero); and
- Analytically bounding for medium and large (with a bound that is better than the bound away from zero in the previous two tasks).

Note that tasks 2 and 3 do not directly require any further understanding of the function .

Below we will give a progress report on the numeric and analytic sides of these tasks.

** — 1. Numerics report (contributed by Sujit Nair) — **

There is some progress on the code side but not at the pace I was hoping. Here are a few things which happened (rather, mistakes which were taken care of).

- We got rid of code which wasn’t being used. For example, @dhjpolymath computed based on an old version but only realized it after the fact.
- We implemented tests to catch human/numerical bugs before a computation starts. Again, we lost some numerical cycles but moving forward these can be avoided.
- David got set up on GitHub and he is able to compare his output (in C) with the Python code. That is helping a lot.

Two areas which were worked on were

- Computing and zeroes for around
- Computing quantities like , , , etc. with the goal of understanding the zero free regions.

Some observations for , , include:

- does seem to avoid the negative real axis
- (based on the oscillations and trends in the plots)
- seems to be settling around range.

See the figure below. The top plot is on the complex plane and the bottom plot is the absolute value. The code to play with this is here.

** — 2. Analysis report — **

The Riemann-Siegel formula and some manipulations (see wiki) give , where

where is a contour that goes from to staying a bounded distance away from the upper imaginary and right real axes, and is the complex conjugate of . (In each of these sums, it is the first term that should dominate, with the second one being about as large.) One can then evolve by the heat flow to obtain , where

Steepest descent heuristics then predict that , , and . For the purposes of this project, we will need effective error estimates here, with explicit error terms.

A start has been made towards this goal at this wiki page. Firstly there is a “effective Laplace method” lemma that gives effective bounds on integrals of the form if the real part of is either monotone with large derivative, or has a critical point and is decreasing on both sides of that critical point. In principle, all one has to do is manipulate expressions such as , , by change of variables, contour shifting and integration by parts until it is of the form to which the above lemma can be profitably applied. As one may imagine though the computations are messy, particularly for the term. As a warm up, I have begun by trying to estimate integrals of the form

for smallish complex numbers , as these sorts of integrals appear in the form of . As of this time of writing, there are effective bounds for the case, and I am currently working on extending them to the case, which should give enough control to approximate and . The most complicated task will be that of upper bounding , but it also looks eventually doable.

This is the second “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this previous thread. 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 now have the following proposition (see this page for a proof sketch) that looks like it can give a numerically feasible approach to bound :

Proposition 1Suppose that one has parameters obeying the following properties:

- All the zeroes of with are real.
- There are no zeroes with in the region .
- There are no zeroes with and .
Then one has .

The first hypothesis is already known for up to about (we should find out exactly what we can reach here). Preliminary calculations suggest that we can obtain the third item provided that . The second hypothesis requires good numerical calculation for , to which we now turn.

The initial definition of is given by the formula

where

This formula has proven numerically computable to acceptable error up until about the first hundred zeroes of , but degrades after that, and so other exact or approximate formulae for are needed. One possible exact formula that could be useful is

where

and

and can be chosen arbitrarily. We are still trying to see if this can be implemented numerically to give better accuracy than the previous formula.

It seems particularly promising to develop a generalisation of the Riemann-Siegel approximate functional equation for . Preliminary computations suggest in particular that we have the approximation

where

Some very preliminary numerics suggest that this formula is reasonably accurate even for moderate values of , though further numerical verification is needed. As a proof of concept, one could take this approximation as exact for the purposes of seeing what ranges of one can feasibly compute with (and for extremely large values of , we will presumably have to introduce some version of the Odlyzko-Schönhage algorithm. Of course, to obtain a rigorous result, we will eventually need a rigorous version of this formula with explicit error bounds. It may also be necessary to add more terms to the approximation to reduce the size of the error.

Sujit Nair has kindly summarised for me the current state of affairs with the numerics as follows:

—

- We need a real milestone and work backward to set up intermediate goals. This will definitely help bring in focus!
- So far, we have some utilities to compute zeroes of with a nonlinear solver which uses roots of as an initial condition. The solver is a wrapper around MINPACK’s implementation of Powell’s method. There is some room for optimization. For example, we aren’t providing the solver with the analytical Jacobian which speeds up the computation and increases accuracy.
- We have some results in the output folder which contains the first 1000 roots of for some small values of , etc. They need some more organization and visualization.

We have a decent initial start but we have some ways to go. Moving forward, here is my proposition for some areas of focus. We should expand and prioritize after some open discussion.

- Short term Optimize the existing framework and target to have the first million zeros of (for a reasonable range of ) and the corresponding plots. With better engineering practice and discipline, I am confident we can get to a few tens of millions range. Some things which will help include parallelization, iterative approaches (using zeroes of to compute zeroes of ), etc.
- Medium term We need to explore better ways to represent the zeros and compute them. An analogy is the computation of Riemann zeroes up to height . It is computed by computing the sign changes of (page 119 of Edwards) and by exploiting the speed up with the Riemann-Siegel formulation (over Euler-Maclaurin). For larger values of , I am not sure the root solver based approach is going to work to understand the gaps between zeroes.
- Long term We also need a better understanding of the errors involved in the computation — truncation, hardware/software, etc.

This is the first official “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant . 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 proposal naturally splits into at least three separate (but loosely related) topics:

- Numerical computation of the entire functions , with the ultimate aim of establishing zero-free regions of the form for various .
- Improved understanding of the dynamics of the zeroes of .
- Establishing the zero-free nature of when and is sufficiently large depending on and .

Below the fold, I will present each of these topics in turn, to initiate further discussion in each of them. (I thought about splitting this post into three to have three separate discussions, but given the current volume of comments, I think we should be able to manage for now having all the comments in a single post. If this changes then of course we can split up some of the discussion later.)

To begin with, let me present some formulae for computing (inspired by similar computations in the Ki-Kim-Lee paper) which may be useful. The initial definition of is

where

is a variant of the Jacobi theta function. We observe that in fact extends analytically to the strip

as has positive real part on this strip. One can use the Poisson summation formula to verify that is even, (see this previous post for details). This lets us obtain a number of other formulae for . Most obviously, one can unfold the integral to obtain

In my previous paper with Brad, we used this representation, combined with Fubini’s theorem to swap the sum and integral, to obtain a useful series representation for in the case because expressions such as diverge as approaches . Nevertheless we can still perform the following contour integration manipulation. Let be fixed. The function decays super-exponentially fast (much faster than , in particular) as with ; as is even, we also have this decay as with (this is despite each of the summands in having much slower decay in this direction – there is considerable cancellation!). Hence by the Cauchy integral formula we have

Splitting the horizontal line from to at and using the even nature of , we thus have

Using the functional equation , we thus have the representation

where is the oscillatory integral

The formula (2) is valid for any . Naively one would think that it would be simplest to take ; however, when and is large (with bounded), it seems asymptotically better to take closer to , in particular something like seems to be a reasonably good choice. This is because the integrand in (3) becomes significantly less oscillatory and also much lower in amplitude; the term in (3) now generates a factor roughly comparable to (which, as we will see below, is the main term in the decay asymptotics for ), while the term still exhibits a reasonable amount of decay as . We will use the representation (2) in the asymptotic analysis of below, but it may also be a useful representation to use for numerical purposes.

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