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This is a sequel to this previous blog post, in which we discussed the effect of the heat flow evolution

on the zeroes of a time-dependent family of polynomials , with a particular focus on the case when the polynomials had real zeroes. Here (inspired by some discussions I had during a recent conference on the Riemann hypothesis in Bristol) we record the analogous theory in which the polynomials instead have zeroes on a circle , with the heat flow slightly adjusted to compensate for this. As we shall discuss shortly, a key example of this situation arises when is the numerator of the zeta function of a curve.

More precisely, let be a natural number. We will say that a polynomial

of degree (so that ) obeys the *functional equation* if the are all real and

for all , thus

and

for all non-zero . This means that the zeroes of (counting multiplicity) lie in and are symmetric with respect to complex conjugation and inversion across the circle . We say that this polynomial *obeys the Riemann hypothesis* if all of its zeroes actually lie on the circle . For instance, in the case, the polynomial obeys the Riemann hypothesis if and only if .

Such polynomials arise in number theory as follows: if is a projective curve of genus over a finite field , then, as famously proven by Weil, the associated local zeta function (as defined for instance in this previous blog post) is known to take the form

where is a degree polynomial obeying both the functional equation and the Riemann hypothesis. In the case that is an elliptic curve, then and takes the form , where is the number of -points of minus . The Riemann hypothesis in this case is a famous result of Hasse.

Another key example of such polynomials arise from rescaled characteristic polynomials

of matrices in the compact symplectic group . These polynomials obey both the functional equation and the Riemann hypothesis. The Sato-Tate conjecture (in higher genus) asserts, roughly speaking, that “typical” polyomials arising from the number theoretic situation above are distributed like the rescaled characteristic polynomials (1), where is drawn uniformly from with Haar measure.

Given a polynomial of degree with coefficients

we can evolve it in time by the formula

thus for . Informally, as one increases , this evolution accentuates the effect of the extreme monomials, particularly, and at the expense of the intermediate monomials such as , and conversely as one decreases . This family of polynomials obeys the heat-type equation

In view of the results of Marcus, Spielman, and Srivastava, it is also very likely that one can interpret this flow in terms of expected characteristic polynomials involving conjugation over the compact symplectic group , and should also be tied to some sort of “” version of Brownian motion on this group, but we have not attempted to work this connection out in detail.

It is clear that if obeys the functional equation, then so does for any other time . Now we investigate the evolution of the zeroes. Suppose at some time that the zeroes of are distinct, then

From the inverse function theorem we see that for times sufficiently close to , the zeroes of continue to be distinct (and vary smoothly in ), with

Differentiating this at any not equal to any of the , we obtain

and

and

Inserting these formulae into (2) (expanding as ) and canceling some terms, we conclude that

for sufficiently close to , and not equal to . Extracting the residue at , we conclude that

which we can rearrange as

If we make the change of variables (noting that one can make depend smoothly on for sufficiently close to ), this becomes

Intuitively, this equation asserts that the phases repel each other if they are real (and attract each other if their difference is imaginary). If obeys the Riemann hypothesis, then the are all real at time , then the Picard uniqueness theorem (applied to and its complex conjugate) then shows that the are also real for sufficiently close to . If we then define the entropy functional

then the above equation becomes a gradient flow

which implies in particular that is non-increasing in time. This shows that as one evolves time forward from , there is a uniform lower bound on the separation between the phases , and hence the equation can be solved indefinitely; in particular, obeys the Riemann hypothesis for all if it does so at time . Our argument here assumed that the zeroes of were simple, but this assumption can be removed by the usual limiting argument.

For any polynomial obeying the functional equation, the rescaled polynomials converge locally uniformly to as . By Rouche’s theorem, we conclude that the zeroes of converge to the equally spaced points on the circle . Together with the symmetry properties of the zeroes, this implies in particular that obeys the Riemann hypothesis for all sufficiently large positive . In the opposite direction, when , the polynomials converge locally uniformly to , so if , of the zeroes converge to the origin and the other converge to infinity. In particular, fails the Riemann hypothesis for sufficiently large negative . Thus (if ), there must exist a real number , which we call the *de Bruijn-Newman constant* of the original polynomial , such that obeys the Riemann hypothesis for and fails the Riemann hypothesis for . The situation is a bit more complicated if vanishes; if is the first natural number such that (or equivalently, ) does not vanish, then by the above arguments one finds in the limit that of the zeroes go to the origin, go to infinity, and the remaining zeroes converge to the equally spaced points . In this case the de Bruijn-Newman constant remains finite except in the degenerate case , in which case .

For instance, consider the case when and for some real with . Then the quadratic polynomial

has zeroes

and one easily checks that these zeroes lie on the circle when , and are on the real axis otherwise. Thus in this case we have (with if ). Note how as increases to , the zeroes repel each other and eventually converge to , while as decreases to , the zeroes collide and then separate on the real axis, with one zero going to the origin and the other to infinity.

The arguments in my paper with Brad Rodgers (discussed in this previous post) indicate that for a “typical” polynomial of degree that obeys the Riemann hypothesis, the expected time to relaxation to equilibrium (in which the zeroes are equally spaced) should be comparable to , basically because the average spacing is and hence by (3) the typical velocity of the zeroes should be comparable to , and the diameter of the unit circle is comparable to , thus requiring time comparable to to reach equilibrium. Taking contrapositives, this suggests that the de Bruijn-Newman constant should typically take on values comparable to (since typically one would not expect the initial configuration of zeroes to be close to evenly spaced). I have not attempted to formalise or prove this claim, but presumably one could do some numerics (perhaps using some of the examples of given previously) to explore this further.

We now approach conformal maps from yet another perspective. Given an open subset of the complex numbers , define a univalent function on to be a holomorphic function that is also injective. We will primarily be studying this concept in the case when is the unit disk .

Clearly, a univalent function on the unit disk is a conformal map from to the image ; in particular, is simply connected, and not all of (since otherwise the inverse map would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset of the complex numbers is the image of a univalent function on . Furthermore, if contains the origin, then the univalent function with this image becomes unique once we normalise and . Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions with and . We will focus particular attention on the univalent functions with the normalisation and ; such functions will be called schlicht functions.

One basic example of a univalent function on is the Cayley transform , which is a Möbius transformation from to the right half-plane . (The slight variant is also referred to as the Cayley transform, as is the closely related map , which maps to the upper half-plane.) One can square this map to obtain a further univalent function , which now maps to the complex numbers with the negative real axis removed. One can normalise this function to be schlicht to obtain the Koebe function

which now maps to the complex numbers with the half-line removed. A little more generally, for any we have the *rotated Koebe function*

that is a schlicht function that maps to the complex numbers with the half-line removed.

Every schlicht function has a convergent Taylor expansion

for some complex coefficients with . For instance, the Koebe function has the expansion

and similarly the rotated Koebe function has the expansion

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients should obey the bound for all . After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions that are *odd*, thus for all , and the Taylor expansion now reads

for some complex coefficients with . One can transform a general schlicht function to an odd schlicht function by observing that the function , after removing the singularity at zero, is a non-zero function that equals at the origin, and thus (as is simply connected) has a unique holomorphic square root that also equals at the origin. If one then sets

it is not difficult to verify that is an odd schlicht function which additionally obeys the equation

Conversely, given an odd schlicht function , the formula (4) uniquely determines a schlicht function .

For instance, if is the Koebe function (1), becomes

which maps to the complex numbers with two slits removed, and if is the rotated Koebe function (2), becomes

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem)Let be a natural number.

- (i) (Robertson conjecture) If is an odd schlicht function, then
- (ii) (Bieberbach conjecture) If is a schlicht function, then

It is easy to see that the Robertson conjecture for a given value of implies the Bieberbach conjecture for the same value of . Indeed, if is schlicht, and is the odd schlicht function given by (3), then from extracting the coefficient of (4) we obtain a formula

for the coefficients of in terms of the coefficients of . Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of from the Robertson conjecture for the same value of . We remark that Littlewood and Paley had conjectured a stronger form of Robertson’s conjecture, but this was disproved for by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of , known as the *Milin conjecture*. Next, one continuously enlarges the image of the schlicht function to cover all of ; done properly, this places the schlicht function as the initial function in a sequence of univalent maps known as a Loewner chain. The functions obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term (or , in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of , , or . After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of ) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

This is the seventh “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , 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.

The most recent news is that we appear to have completed the verification that is free of zeroes when and , which implies that . For very large (for instance when the quantity is at least ) this can be done analytically; for medium values of (say when is between and ) this can be done by numerically evaluating a fast approximation to and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of , in part due to some new, and significantly faster, numerical ways to evaluate in this range.

One obvious thing to do now is to experiment with lowering the parameters and and see what happens. However there are two other potential ways to bound which may also be numerically feasible. One approach is based on trying to exclude zeroes of in a region of the form , and for some moderately large (this acts as a “barrier” to prevent zeroes from flowing into the region at time , assuming that they were not already there at time ). This require significantly less numerical verification in the aspect, but more numerical verification in the aspect, so it is not yet clear whether this is a net win.

Another, rather different approach, is to study the evolution of statistics such as over time. One has fairly good control on such quantities at time zero, and their time derivative looks somewhat manageable, so one may be able to still have good control on this quantity at later times . However for this approach to work, one needs an effective version of the Riemann-von Mangoldt formula for , which at present is only available asymptotically (or at time ). This approach may be able to avoid almost all numerical computation, except for numerical verification of the Riemann hypothesis, for which we can appeal to existing literature.

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

This is the sixth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , 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.

The last two threads have been focused primarily on the test problem of showing that whenever . We have been able to prove this for most regimes of , or equivalently for most regimes of the natural number parameter . In many of these regimes, a certain explicit approximation to was used, together with a non-zero normalising factor ; see the wiki for definitions. The explicit upper bound

has been proven for certain explicit expressions (see here) depending on . In particular, if satisfies the inequality

then is non-vanishing thanks to the triangle inequality. (In principle we have an even more accurate approximation available, but it is looking like we will not need it for this test problem at least.)

We have explicit upper bounds on , , ; see this wiki page for details. They are tabulated in the range here. For , the upper bound for is monotone decreasing, and is in particular bounded by , while and are known to be bounded by and respectively (see here).

Meanwhile, the quantity can be lower bounded by

for certain explicit coefficients and an explicit complex number . Using the triangle inequality to lower bound this by

we can obtain a lower bound of for , which settles the test problem in this regime. One can get more efficient lower bounds by multiplying both Dirichlet series by a suitable Euler product mollifier; we have found for to be good choices to get a variety of further lower bounds depending only on , see this table and this wiki page. Comparing this against our tabulated upper bounds for the error terms we can handle the range .

In the range , we have been able to obtain a suitable lower bound (where exceeds the upper bound for ) by numerically evaluating at a mesh of points for each choice of , with the mesh spacing being adaptive and determined by and an upper bound for the derivative of ; the data is available here.

This leaves the final range (roughly corresponding to ). Here we can numerically evaluate to high accuracy at a fine mesh (see the data here), but to fill in the mesh we need good upper bounds on . It seems that we can get reasonable estimates using some contour shifting from the original definition of (see here). We are close to finishing off this remaining region and thus solving the toy problem.

Beyond this, we need to figure out how to show that for as well. General theory lets one do this for , leaving the region . The analytic theory that handles and should also handle this region; for presumably the argument principle will become relevant.

The full argument also needs to be streamlined and organised; right now it sprawls over many wiki pages and github code files. (A very preliminary writeup attempt has begun here). We should also see if there is much hope of extending the methods to push much beyond the bound of that we would get from the above calculations. This would also be a good time to start discussing whether to move to the writing phase of the project, or whether there are still fruitful research directions for the project to explore.

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

This is the fifth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , 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 whenever . We have two useful approximations for , which we have denoted and , and a normalising quantity that is asymptotically equal to the above expressions; see the wiki page for definitions. In practice, the approximation seems to be accurate within about one or two significant figures, whilst the approximation is accurate to about three or four. We have an effective upper bound

where the expressions are quite small in practice ( is typically about two orders of magnitude smaller than the main term once is moderately large, and the error terms are even smaller). See this page for details. In principle we could also obtain an effective upper bound for (the term would be replaced by something smaller).

The ratio takes the form of a difference of two Dirichlet series, where is a phase whose value is explicit but perhaps not terribly important, and the coefficients are explicit and relatively simple ( is , and is approximately ). To bound this away from zero, we have found it advantageous to mollify this difference by multiplying by an Euler product to cancel much of the initial oscillation; also one can take advantage of the fact that the are real and the 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 or so. The error terms are already quite small by this stage, so we should soon be able to rigorously keep 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 become unmanageable. For very small we may have to explore some faster ways to compute the expression , which is still difficult to compute directly with high accuracy. One will also need to bound the somewhat unwieldy expressions by something more manageable. For instance, right now these quantities depend on the continuous variable ; it would be preferable to have a quantity that depends only on the parameter , as this could be computed numerically for all 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.

This is the fourth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing https://terrytao.wordpress.com/2018/01/24/polymath-proposal-upper-bounding-the-de-bruijn-newman-constant/. Progress will be summarised at this Polymath wiki page.

We are getting closer to finishing off the following test problem: can one show that whenever , ? This would morally show that . A wiki page for this problem has now been created here. We have obtained a number of approximations to (see wiki page), though numeric evidence indicates that the approximations are all very close to each other. (Many of these approximations come with a correction term , but thus far it seems that we may be able to avoid having to use this refinement to the approximations.) The effective approximation also comes with an effective error bound

for some explicit (but somewhat messy) error terms : see this wiki page for details. The original approximations can be considered deprecated at this point in favour of the (slightly more complicated) approximation ; the approximation is a simplified version of which is not quite as accurate but might be useful for testing purposes.

It is convenient to normalise everything by an explicit non-zero factor . Asymptotically, converges to 1; numerically, it appears that its magnitude (and also its real part) stays roughly between 0.4 and 3 in the range , and we seem to be able to keep it (or at least the toy counterpart ) away from zero starting from about (here it seems that there is a useful trick of multiplying by Euler-type factors like to cancel off some of the oscillation). Also, the bounds on the error seem to be of size about 0.1 or better in these ranges also. So we seem to be on track to be able to rigorously eliminate zeroes starting from about or so. We have not discussed too much what to do with the small values of ; at some point our effective error bounds will become unusable, and we may have to find some more faster ways to compute .

In addition to this main direction of inquiry, there have been additional discussions on the dynamics of zeroes, and some numerical investigations of the behaviour Lehmer pairs under heat flow. Contributors are welcome to summarise any findings from these discussions from previous threads (or on any other related topic, e.g. improvements in the code) in the comments below.

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. Unfortunately this is not possible 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.

Brad Rodgers and I have uploaded to the arXiv our paper “The De Bruijn-Newman constant is non-negative“. This paper affirms a conjecture of Newman regarding to the extent to which the Riemann hypothesis, if true, is only “barely so”. To describe the conjecture, let us begin with the Riemann xi function

where is the Gamma function and is the Riemann zeta function. Initially, this function is only defined for , but, as was already known to Riemann, we can manipulate it into a form that extends to the entire complex plane as follows. Firstly, in view of the standard identity , we can write

and hence

By a rescaling, one may write

and similarly

and thus (after applying Fubini’s theorem)

We’ll make the change of variables to obtain

If we introduce the mild renormalisation

of , we then conclude (at least for ) that

which one can verify to be rapidly decreasing both as and as , with the decrease as faster than any exponential. In particular extends holomorphically to the upper half plane.

If we normalize the Fourier transform of a (Schwartz) function as , it is well known that the Gaussian is its own Fourier transform. The creation operator interacts with the Fourier transform by the identity

Since , this implies that the function

is its own Fourier transform. (One can view the polynomial as a renormalised version of the fourth Hermite polynomial.) Taking a suitable linear combination of this with , we conclude that

is also its own Fourier transform. Rescaling by and then multiplying by , we conclude that the Fourier transform of

is

and hence by the Poisson summation formula (using symmetry and vanishing at to unfold the summation in (2) to the integers rather than the natural numbers) we obtain the functional equation

which implies that and are even functions (in particular, now extends to an entire function). From this symmetry we can also rewrite (1) as

which now gives a convergent expression for the entire function for all complex . As is even and real-valued on , is even and also obeys the functional equation , which is equivalent to the usual functional equation for the Riemann zeta function. The Riemann hypothesis is equivalent to the claim that all the zeroes of are real.

De Bruijn introduced the family of deformations of , defined for all and by the formula

From a PDE perspective, one can view as the evolution of under the backwards heat equation . As with , the are all even entire functions that obey the functional equation , and one can ask an analogue of the Riemann hypothesis for each such , namely whether all the zeroes of are real. De Bruijn showed that these hypotheses were monotone in : if had all real zeroes for some , then would also have all zeroes real for any . Newman later sharpened this claim by showing the existence of a finite number , now known as the *de Bruijn-Newman constant*, with the property that had all zeroes real if and only if . Thus, the Riemann hypothesis is equivalent to the inequality . Newman then conjectured the complementary bound ; in his words, this conjecture asserted that if the Riemann hypothesis is true, then it is only “barely so”, in that the reality of all the zeroes is destroyed by applying heat flow for even an arbitrarily small amount of time. Over time, a significant amount of evidence was established in favour of this conjecture; most recently, in 2011, Saouter, Gourdon, and Demichel showed that .

In this paper we finish off the proof of Newman’s conjecture, that is we show that . The proof is by contradiction, assuming that (which among other things, implies the truth of the Riemann hypothesis), and using the properties of backwards heat evolution to reach a contradiction.

Very roughly, the argument proceeds as follows. As observed by Csordas, Smith, and Varga (and also discussed in this previous blog post, the backwards heat evolution of the introduces a nice ODE dynamics on the zeroes of , namely that they solve the ODE

for all (one has to interpret the sum in a principal value sense as it is not absolutely convergent, but let us ignore this technicality for the current discussion). Intuitively, this ODE is asserting that the zeroes repel each other, somewhat like positively charged particles (but note that the dynamics is first-order, as opposed to the second-order laws of Newtonian mechanics). Formally, a steady state (or equilibrium) of this dynamics is reached when the are arranged in an arithmetic progression. (Note for instance that for any positive , the functions obey the same backwards heat equation as , and their zeroes are on a fixed arithmetic progression .) The strategy is to then show that the dynamics from time to time creates a *convergence to local equilibrium*, in which the zeroes locally resemble an arithmetic progression at time . This will be in contradiction with known results on pair correlation of zeroes (or on related statistics, such as the fluctuations on gaps between zeroes), such as the results of Montgomery (actually for technical reasons it is slightly more convenient for us to use related results of Conrey, Ghosh, Goldston, Gonek, and Heath-Brown). Another way of thinking about this is that even very slight deviations from local equilibrium (such as a small number of gaps that are slightly smaller than the average spacing) will almost immediately lead to zeroes colliding with each other and leaving the real line as one evolves backwards in time (i.e., under the *forward* heat flow). This is a refinement of the strategy used in previous lower bounds on , in which “Lehmer pairs” (pairs of zeroes of the zeta function that were unusually close to each other) were used to limit the extent to which the evolution continued backwards in time while keeping all zeroes real.

How does one obtain this convergence to local equilibrium? We proceed by broad analogy with the “local relaxation flow” method of Erdos, Schlein, and Yau in random matrix theory, in which one combines some initial control on zeroes (which, in the case of the Erdos-Schlein-Yau method, is referred to with terms such as “local semicircular law”) with convexity properties of a relevant Hamiltonian that can be used to force the zeroes towards equilibrium.

We first discuss the initial control on zeroes. For , we have the classical Riemann-von Mangoldt formula, which asserts that the number of zeroes in the interval is as . (We have a factor of here instead of the more familiar due to the way is normalised.) This implies for instance that for a fixed , the number of zeroes in the interval is . Actually, because we get to assume the Riemann hypothesis, we can sharpen this to , a result of Littlewood (see this previous blog post for a proof). Ideally, we would like to obtain similar control for the other , , as well. Unfortunately we were only able to obtain the weaker claims that the number of zeroes of in is , and that the number of zeroes in is , that is to say we only get good control on the distribution of zeroes at scales rather than at scales . Ultimately this is because we were only able to get control (and in particular, lower bounds) on with high precision when (whereas has good estimates as soon as is larger than (say) ). This control is obtained by the expressing in terms of some contour integrals and using the method of steepest descent (actually it is slightly simpler to rely instead on the Stirling approximation for the Gamma function, which can be proven in turn by steepest descent methods). Fortunately, it turns out that this weaker control is still (barely) enough for the rest of our argument to go through.

Once one has the initial control on zeroes, we now need to force convergence to local equilibrium by exploiting convexity of a Hamiltonian. Here, the relevant Hamiltonian is

ignoring for now the rather important technical issue that this sum is not actually absolutely convergent. (Because of this, we will need to truncate and renormalise the Hamiltonian in a number of ways which we will not detail here.) The ODE (3) is formally the gradient flow for this Hamiltonian. Furthermore, this Hamiltonian is a convex function of the (because is a convex function on ). We therefore expect the Hamiltonian to be a decreasing function of time, and that the derivative should be an increasing function of time. As time passes, the derivative of the Hamiltonian would then be expected to converge to zero, which should imply convergence to local equilibrium.

Formally, the derivative of the above Hamiltonian is

Again, there is the important technical issue that this quantity is infinite; but it turns out that if we renormalise the Hamiltonian appropriately, then the energy will also become suitably renormalised, and in particular will vanish when the are arranged in an arithmetic progression, and be positive otherwise. One can also formally calculate the derivative of to be a somewhat complicated but manifestly non-negative quantity (a sum of squares); see this previous blog post for analogous computations in the case of heat flow on polynomials. After flowing from time to time , and using some crude initial bounds on and in this region (coming from the Riemann-von Mangoldt type formulae mentioned above and some further manipulations), we can eventually show that the (renormalisation of the) energy at time zero is small, which forces the to locally resemble an arithmetic progression, which gives the required convergence to local equilibrium.

There are a number of technicalities involved in making the above sketch of argument rigorous (for instance, justifying interchanges of derivatives and infinite sums turns out to be a little bit delicate). I will highlight here one particular technical point. One of the ways in which we make expressions such as the energy finite is to truncate the indices to an interval to create a truncated energy . In typical situations, we would then expect to be decreasing, which will greatly help in bounding (in particular it would allow one to control by time-averaged quantities such as , which can in turn be controlled using variants of (4)). However, there are boundary effects at both ends of that could in principle add a large amount of energy into , which is bad news as it could conceivably make undesirably large even if integrated energies such as remain adequately controlled. As it turns out, such boundary effects are negligible as long as there is a large gap between adjacent zeroes at boundary of – it is only narrow gaps that can rapidly transmit energy across the boundary of . Now, narrow gaps can certainly exist (indeed, the GUE hypothesis predicts these happen a positive fraction of the time); but the pigeonhole principle (together with the Riemann-von Mangoldt formula) can allow us to pick the endpoints of the interval so that no narrow gaps appear at the boundary of for any given time . However, there was a technical problem: this argument did not allow one to find a single interval that avoided gaps for *all* times simultaneously – the pigeonhole principle could produce a different interval for each time ! Since the number of times was uncountable, this was a serious issue. (In physical terms, the problem was that there might be very fast “longitudinal waves” in the dynamics that, at each time, cause some gaps between zeroes to be highly compressed, but the specific gap that was narrow changed very rapidly with time. Such waves could, in principle, import a huge amount of energy into by time .) To resolve this, we borrowed a PDE trick of Bourgain’s, in which the pigeonhole principle was coupled with local conservation laws. More specifically, we use the phenomenon that very narrow gaps take a nontrivial amount of time to expand back to a reasonable size (this can be seen by comparing the evolution of this gap with solutions of the scalar ODE , which represents the fastest at which a gap such as can expand). Thus, if a gap is reasonably large at some time , it will also stay reasonably large at slightly earlier times for some moderately small . This lets one locate an interval that has manageable boundary effects during the times in , so in particular is basically non-increasing in this time interval. Unfortunately, this interval is a little bit too short to cover all of ; however it turns out that one can iterate the above construction and find a nested sequence of intervals , with each non-increasing in a different time interval , and with all of the time intervals covering . This turns out to be enough (together with the obvious fact that is monotone in ) to still control for some reasonably sized interval , as required for the rest of the arguments.

ADDED LATER: the following analogy (involving functions with just two zeroes, rather than an infinite number of zeroes) may help clarify the relation between this result and the Riemann hypothesis (and in particular why this result does not make the Riemann hypothesis any easier to prove, in fact it confirms the delicate nature of that hypothesis). Suppose one had a quadratic polynomial of the form , where was an unknown real constant. Suppose that one was for some reason interested in the analogue of the “Riemann hypothesis” for , namely that all the zeroes of are real. A priori, there are three scenarios:

- (Riemann hypothesis false) , and has zeroes off the real axis.
- (Riemann hypothesis true, but barely so) , and both zeroes of are on the real axis; however, any slight perturbation of in the positive direction would move zeroes off the real axis.
- (Riemann hypothesis true, with room to spare) , and both zeroes of are on the real axis. Furthermore, any slight perturbation of will also have both zeroes on the real axis.

The analogue of our result in this case is that , thus ruling out the third of the three scenarios here. In this simple example in which only two zeroes are involved, one can think of the inequality as asserting that if the zeroes of are real, then they must be repeated. In our result (in which there are an infinity of zeroes, that become increasingly dense near infinity), and in view of the convergence to local equilibrium properties of (3), the analogous assertion is that if the zeroes of are real, then they do not behave locally as if they were in arithmetic progression.

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