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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.

This is the seventh “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.

The most recent news is that we appear to have completed the verification that {H_t(x+iy)} is free of zeroes when {t=0.4} and {y \geq 0.4}, which implies that {\Lambda \leq 0.48}. For very large {x} (for instance when the quantity {N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor} is at least {300}) this can be done analytically; for medium values of {x} (say when {N} is between {11} and {300}) this can be done by numerically evaluating a fast approximation {A^{eff} + B^{eff}} to {H_t} and using the argument principle in a rectangle; and most recently it appears that we can also handle small values of {x}, in part due to some new, and significantly faster, numerical ways to evaluate {H_t} in this range.

One obvious thing to do now is to experiment with lowering the parameters {t} and {y} and see what happens. However there are two other potential ways to bound {\Lambda} which may also be numerically feasible. One approach is based on trying to exclude zeroes of {H_t(x+iy)=0} in a region of the form {0 \leq t \leq t_0}, {X \leq x \leq X+1} and {y \geq y_0} for some moderately large {X} (this acts as a “barrier” to prevent zeroes from flowing into the region {\{ 0 \leq x \leq X, y \geq y_0 \}} at time {t_0}, assuming that they were not already there at time {0}). This require significantly less numerical verification in the {x} aspect, but more numerical verification in the {t} 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 {S(t) = \sum_{H_t(x+iy)=0: x,y>0} y e^{-x/X}} 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 {t_0>0}. However for this approach to work, one needs an effective version of the Riemann-von Mangoldt formula for {H_t}, which at present is only available asymptotically (or at time {t=0}). 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 {\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.

The last two threads have been focused primarily on the test problem of showing that {H_t(x+iy) \neq 0} whenever {t = y = 0.4}. We have been able to prove this for most regimes of {x}, or equivalently for most regimes of the natural number parameter {N := \lfloor \sqrt{\frac{x}{4\pi} + \frac{t}{16}} \rfloor}. In many of these regimes, a certain explicit approximation {A^{eff}+B^{eff}} to {H_t} was used, together with a non-zero normalising factor {B^{eff}_0}; see the wiki for definitions. The explicit upper bound

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

has been proven for certain explicit expressions {E_1, E_2, E_3} (see here) depending on {x}. In particular, if {x} satisfies the inequality

\displaystyle  |\frac{A^{eff}+B^{eff}}{B^{eff}_0}| > \frac{E_1}{|B^{eff}_0|} + \frac{E_2}{|B^{eff}_0|} + \frac{E_3}{|B^{eff}_0|}

then {H_t(x+iy)} is non-vanishing thanks to the triangle inequality. (In principle we have an even more accurate approximation {A^{eff}+B^{eff}-C^{eff}} available, but it is looking like we will not need it for this test problem at least.)

We have explicit upper bounds on {\frac{E_1}{|B^{eff}_0|}}, {\frac{E_2}{|B^{eff}_0|}}, {\frac{E_3}{|B^{eff}_0|}}; see this wiki page for details. They are tabulated in the range {3 \leq N \leq 2000} here. For {N \geq 2000}, the upper bound {\frac{E_3^*}{|B^{eff}_0|}} for {\frac{E_3}{|B^{eff}_0|}} is monotone decreasing, and is in particular bounded by {1.53 \times 10^{-5}}, while {\frac{E_2}{|B^{eff}_0|}} and {\frac{E_1}{|B^{eff}_0|}} are known to be bounded by {2.9 \times 10^{-7}} and {2.8 \times 10^{-8}} respectively (see here).

Meanwhile, the quantity {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|} can be lower bounded by

\displaystyle  |\sum_{n=1}^N \frac{b_n}{n^s}| - |\sum_{n=1}^N \frac{a_n}{n^s}|

for certain explicit coefficients {a_n,b_n} and an explicit complex number {s = \sigma + i\tau}. Using the triangle inequality to lower bound this by

\displaystyle  |b_1| - \sum_{n=2}^N \frac{|b_n|}{n^\sigma} - \sum_{n=1}^N \frac{|a_n|}{n^\sigma}

we can obtain a lower bound of {0.18} for {N \geq 2000}, 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 {\prod_{p \leq P} (1 - \frac{b_p}{p^s})} for {P=2,3,5,7} to be good choices to get a variety of further lower bounds depending only on {N}, see this table and this wiki page. Comparing this against our tabulated upper bounds for the error terms we can handle the range {300 \leq N \leq 2000}.

In the range {11 \leq N \leq 300}, we have been able to obtain a suitable lower bound {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}| \geq c} (where {c} exceeds the upper bound for {\frac{E_1}{|B^{eff}_0|} + \frac{E_2}{|B^{eff}_0|} + \frac{E_3}{|B^{eff}_0|}}) by numerically evaluating {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|} at a mesh of points for each choice of {N}, with the mesh spacing being adaptive and determined by {c} and an upper bound for the derivative of {|\frac{A^{eff}+B^{eff}}{B^{eff}_0}|}; the data is available here.

This leaves the final range {N \leq 10} (roughly corresponding to {x \leq 1600}). Here we can numerically evaluate {H_t(x+iy)} to high accuracy at a fine mesh (see the data here), but to fill in the mesh we need good upper bounds on {H'_t(x+iy)}. It seems that we can get reasonable estimates using some contour shifting from the original definition of {H_t} (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 {H_t(x+iy) \neq 0} for {y > 0.4} as well. General theory lets one do this for {y \geq \sqrt{1-2t} = 0.447\dots}, leaving the region {0.4 < y < 0.448}. The analytic theory that handles {N \geq 2000} and {300 \leq N \leq 2000} should also handle this region; for {N \leq 300} 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 {\Lambda \leq 0.48} 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 {\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.

This is the fourth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, 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 {H_t(x+iy) \neq 0} whenever {t = y = 0.4}, {x \geq 0}? This would morally show that {\Lambda \leq 0.48}. A wiki page for this problem has now been created here. We have obtained a number of approximations {A+B, A'+B', A^{eff}+B^{eff}, A^{toy}+B^{toy}} to {H_t} (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 {C}, but thus far it seems that we may be able to avoid having to use this refinement to the approximations.) The effective approximation {A^{eff} + B^{eff}} also comes with an effective error bound

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

for some explicit (but somewhat messy) error terms {E_1,E_2,E_3}: see this wiki page for details. The original approximations {A+B, A'+B'} can be considered deprecated at this point in favour of the (slightly more complicated) approximation {A^{eff}+B^{eff}}; the approximation {A^{toy}+B^{toy}} is a simplified version of {A^{eff}+B^{eff}} 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 {B^{eff}_0}. Asymptotically, {(A^{eff} + B^{eff}) / B^{eff}_0} converges to 1; numerically, it appears that its magnitude (and also its real part) stays roughly between 0.4 and 3 in the range {10^5 \leq x \leq 10^6}, and we seem to be able to keep it (or at least the toy counterpart {(A^{toy} + B^{toy}) / B^{toy}_0}) away from zero starting from about {x \geq 4 \times 10^6} (here it seems that there is a useful trick of multiplying by Euler-type factors like {1 - \frac{1}{2^{1-s}}} to cancel off some of the oscillation). Also, the bounds on the error {(H_t - A^{eff} - B^{eff}) / B^{eff}_0} 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 {x \geq 10^5} or so. We have not discussed too much what to do with the small values of {x}; at some point our effective error bounds will become unusable, and we may have to find some more faster ways to compute {H_t}.

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.

Kevin Ford, Sergei Konyagin, James Maynard, Carl Pomerance, and I have uploaded to the arXiv our paper “Long gaps in sieved sets“, submitted to J. Europ. Math. Soc..

This paper originated from the MSRI program in analytic number theory last year, and was centred around variants of the question of finding large gaps between primes. As discussed for instance in this previous post, it is now known that within the set of primes {{\mathcal P} = \{2,3,5,\dots\}}, one can find infinitely many adjacent elements {a,b} whose gap {b-a} obeys a lower bound of the form

\displaystyle b-a \gg \log a \frac{\log_2 a \log_4 a}{\log_3 a}

where {\log_k} denotes the {k}-fold iterated logarithm. This compares with the trivial bound of {b-a \gg \log a} that one can obtain from the prime number theorem and the pigeonhole principle. Several years ago, Pomerance posed the question of whether analogous improvements to the trivial bound can be obtained for such sets as

\displaystyle {\mathcal P}_2 = \{ n \in {\bf N}: n^2+1 \hbox{ prime} \}.

Here there is the obvious initial issue that this set is not even known to be infinite (this is the fourth Landau problem), but let us assume for the sake of discussion that this set is indeed infinite, so that we have an infinite number of gaps to speak of. Standard sieve theory techniques give upper bounds for the density of {{\mathcal P}_2} that is comparable (up to an absolute constant) to the prime number theorem bounds for {{\mathcal P}}, so again we can obtain a trivial bound of {b-a \gg \log a} for the gaps of {{\mathcal P}_2}. In this paper we improve this to

\displaystyle b-a \gg \log a \log^c_2 a

for an absolute constant {c>0}; this is not as strong as the corresponding bound for {{\mathcal P}}, but still improves over the trivial bound. In fact we can handle more general “sifted sets” than just {{\mathcal P}_2}. Recall from the sieve of Eratosthenes that the elements of {{\mathcal P}} in, say, the interval {[x/2, x]} can be obtained by removing from {[x/2, x]} one residue class modulo {p} for each prime up to {\sqrt{x}}, namely the class {0} mod {p}. In a similar vein, the elements of {{\mathcal P}_2} in {[x/2,x]} can be obtained by removing for each prime {p} up to {x} zero, one, or two residue classes modulo {p}, depending on whether {-1} is a quadratic residue modulo {p}. On the average, one residue class will be removed (this is a very basic case of the Chebotarev density theorem), so this sieving system is “one-dimensional on the average”. Roughly speaking, our arguments apply to any other set of numbers arising from a sieving system that is one-dimensional on average. (One can consider other dimensions also, but unfortunately our methods seem to give results that are worse than a trivial bound when the dimension is less than or greater than one.)

The standard “Erdős-Rankin” method for constructing long gaps between primes proceeds by trying to line up some residue classes modulo small primes {p} so that they collectively occupy a long interval. A key tool in doing so are the smooth number estimates of de Bruijn and others, which among other things assert that if one removes from an interval such as {[1,x]} all the residue classes {0} mod {p} for {p} between {x^{1/u}} and {x} for some fixed {u>1}, then the set of survivors has exceptionally small density (roughly of the order of {u^{-u}}, with the precise density given by the Dickman function), in marked contrast to the situation in which one randomly removes one residue class for each such prime {p}, in which the density is more like {1/u}. One generally exploits this phenomenon to sieve out almost all the elements of a long interval using some of the primes available, and then using the remaining primes to cover up the remaining elements that have not already been sifted out. In the more recent work on this problem, advanced combinatorial tools such as hypergraph covering lemmas are used for the latter task.

In the case of {{\mathcal P}_2}, there does not appear to be any analogue of smooth numbers, in the sense that there is no obvious way to arrange the residue classes so that they have significantly fewer survivors than a random arrangement. Instead we adopt the following semi-random strategy to cover an interval {[1,y]} by residue classes. Firstly, we randomly remove residue classes for primes {p} up to some intermediate threshold {z} (smaller than {y} by a logarithmic factor), leaving behind a preliminary sifted set {S_{[2,z]}}. Then, for each prime {p} between {z} and another intermediate threshold {x/2}, we remove a residue class mod {p} that maximises (or nearly maximises) its intersection with {S_{[2,z]}}. This ends up reducing the number of survivors to be significantly below what one would achieve if one selects residue classes randomly, particularly if one also uses the hypergraph covering lemma from our previous paper. Finally, we cover each the remaining survivors by a residue class from a remaining available prime.

This is the second “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, 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 {\Lambda}:

Proposition 1 Suppose that one has parameters {t_0, T, \varepsilon > 0} obeying the following properties:

  • All the zeroes of {H_0(x+iy)=0} with {0 \leq x \leq T} are real.
  • There are no zeroes {H_t(x+iy)=0} with {0 \leq t \leq t_0} in the region {\{ x+iy: x \geq T; 1-2t \geq y^2 \geq \varepsilon^2 + (T-x)^2 \}}.
  • There are no zeroes {H_{t_0}(x+iy)=0} with {x > T} and {y \geq \varepsilon}.

Then one has {\Lambda \leq t_0 + \frac{1}{2} \varepsilon^2}.

The first hypothesis is already known for {T} up to about {10^{12}} (we should find out exactly what we can reach here). Preliminary calculations suggest that we can obtain the third item provided that {t_0, \varepsilon \gg \frac{1}{\log T}}. The second hypothesis requires good numerical calculation for {H_t}, to which we now turn.

The initial definition of {H_t} is given by the formula

\displaystyle  H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du

where

\displaystyle  \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u} ) \exp(-\pi n^2 e^{4u}).

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

\displaystyle  H_t(z) = \frac{1}{2} (K_{t,\theta}(z) + \overline{K_{t,\theta}(\overline{z})})

where

\displaystyle  K_{t,\theta}(z) := \sum_{n=1}^\infty (2\pi^2 n^4 I_{t,\theta}(z-9i, \pi n^2) - 3\pi n^2I_{t,\theta}(z-5i, \pi n^2))

and

\displaystyle  I_{t,\theta}(b,\beta) := \int_{i\theta}^{i\theta+i\infty} \exp(tu^2 - \beta e^{4u} + ibu)\ du

and {-\pi/8 < \theta < \pi/8} 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 {H_0}. Preliminary computations suggest in particular that we have the approximation

\displaystyle  H_t(x+iy) \approx \frac{1}{4} (F_t(\frac{1+ix-y}{2}) + \overline{F_t(\frac{1+ix+y}{2})})

where

\displaystyle  F_t(s) := \pi^{-s/2} \Gamma(\frac{s+4}{2}) \sum_{n \leq \sqrt{\mathrm{Im}(s)/2\pi}} \frac{\exp( \frac{t}{16} \log^2 \frac{s+4}{2\pi n^2})}{n^s}.

Some very preliminary numerics suggest that this formula is reasonably accurate even for moderate values of {x}, 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 {T} one can feasibly compute with (and for extremely large values of {T}, 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 {H_t} with a nonlinear solver which uses roots of {H_0} 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 {H_t} for some small values of {t \in \{0.01, 0.1, 0.22\}}, 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.

  1. Short term Optimize the existing framework and target to have the first million zeros of {H_t} (for a reasonable range of {t}) 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 {H_t} to compute zeroes of {H_{t + \delta t}}), etc.
  2. 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 {T}. It is computed by computing the sign changes of {Z(t)} (page 119 of Edwards) and by exploiting the {\sqrt T} speed up with the Riemann-Siegel formulation (over Euler-Maclaurin). For larger values of {j}, I am not sure the root solver based approach is going to work to understand the gaps between zeroes.
  3. 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 {\Lambda}. 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 {H_t(z)}, with the ultimate aim of establishing zero-free regions of the form {\{ x+iy: 0 \leq x \leq T, y \geq \varepsilon \}} for various {T, \varepsilon > 0}.
  • Improved understanding of the dynamics of the zeroes {z_j(t)} of {H_t}.
  • Establishing the zero-free nature of {H_t(x+iy)} when {y \geq \varepsilon > 0} and {x} is sufficiently large depending on {t} and {\varepsilon}.

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 {H_t} (inspired by similar computations in the Ki-Kim-Lee paper) which may be useful. The initial definition of {H_t} is

\displaystyle  H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du

where

\displaystyle  \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(- \pi n^2 e^{4u} )

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

\displaystyle  \{ u \in {\bf C}: -\frac{\pi}{8} < \mathrm{Im} u < \frac{\pi}{8} \}, \ \ \ \ \ (1)

as {e^{4u}} has positive real part on this strip. One can use the Poisson summation formula to verify that {\Phi} is even, {\Phi(-u) = \Phi(u)} (see this previous post for details). This lets us obtain a number of other formulae for {H_t}. Most obviously, one can unfold the integral to obtain

\displaystyle  H_t(z) = \frac{1}{2} \int_{-\infty}^\infty e^{tu^2} \Phi(u) e^{izu}\ du.

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 {H_t} in the {t<0} case. Unfortunately this is not possible in the {t>0} case because expressions such as {e^{tu^2} e^{9u} \exp( -\pi n^2 e^{4u} ) e^{izu}} diverge as {u} approaches {-\infty}. Nevertheless we can still perform the following contour integration manipulation. Let {0 \leq \theta < \frac{\pi}{8}} be fixed. The function {\Phi} decays super-exponentially fast (much faster than {e^{tu^2}}, in particular) as {\mathrm{Re} u \rightarrow +\infty} with {-\infty \leq \mathrm{Im} u \leq \theta}; as {\Phi} is even, we also have this decay as {\mathrm{Re} u \rightarrow -\infty} with {-\infty \leq \mathrm{Im} u \leq \theta} (this is despite each of the summands in {\Phi} having much slower decay in this direction – there is considerable cancellation!). Hence by the Cauchy integral formula we have

\displaystyle  H_t(z) = \frac{1}{2} \int_{i\theta-\infty}^{i\theta+\infty} e^{tu^2} \Phi(u) e^{izu}\ du.

Splitting the horizontal line from {i\theta-\infty} to {i\theta+\infty} at {i\theta} and using the even nature of {\Phi(u)}, we thus have

\displaystyle  H_t(z) = \frac{1}{2} ( \int_{i\theta}^{i\theta+\infty} e^{tu^2} \Phi(u) e^{izu}\ du + \int_{-i\theta}^{-i\theta+\infty} e^{tu^2} \Phi(u) e^{-izu}\ du.

Using the functional equation {\Phi(\overline{u}) = \overline{\Phi(u)}}, we thus have the representation

\displaystyle  H_t(z) = \frac{1}{2} ( K_{t,\theta}(z) + \overline{K_{t,\theta}(\overline{z})} ) \ \ \ \ \ (2)

where

\displaystyle  K_{t,\theta}(z) := \int_{i\theta}^{i \theta+\infty} e^{tu^2} \Phi(u) e^{izu}\ du

\displaystyle  = \sum_{n=1}^\infty 2 \pi^2 n^4 I_{t, \theta}( z - 9i, \pi n^2 ) - 3 \pi n^2 I_{t,\theta}( z - 5i, \pi n^2 )

where {I_{t,\theta}(b,\beta)} is the oscillatory integral

\displaystyle  I_{t,\theta}(b,\beta) := \int_{i\theta}^{i\theta+\infty} \exp( tu^2 - \beta e^{4u} + i b u )\ du. \ \ \ \ \ (3)

The formula (2) is valid for any {0 \leq \theta < \frac{\pi}{8}}. Naively one would think that it would be simplest to take {\theta=0}; however, when {z=x+iy} and {x} is large (with {y} bounded), it seems asymptotically better to take {\theta} closer to {\pi/8}, in particular something like {\theta = \frac{\pi}{8} - \frac{1}{4x}} 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 {\exp(ibu)} term in (3) now generates a factor roughly comparable to {\exp( - \pi x/8 )} (which, as we will see below, is the main term in the decay asymptotics for {H_t(x+iy)}), while the {\exp( - \beta e^{4u} )} term still exhibits a reasonable amount of decay as {u \rightarrow \infty}. We will use the representation (2) in the asymptotic analysis of {H_t} below, but it may also be a useful representation to use for numerical purposes.

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Building on the interest expressed in the comments to this previous post, I am now formally proposing to initiate a “Polymath project” on the topic of obtaining new upper bounds on the de Bruijn-Newman constant {\Lambda}. The purpose of this post is to describe the proposal and discuss the scope and parameters of the project.

De Bruijn introduced a family {H_t: {\bf C} \rightarrow {\bf C}} of entire functions for each real number {t}, defined by the formula

\displaystyle H_t(z) := \int_0^\infty e^{tu^2} \Phi(u) \cos(zu)\ du

where {\Phi} is the super-exponentially decaying function

\displaystyle \Phi(u) := \sum_{n=1}^\infty (2\pi^2 n^4 e^{9u} - 3 \pi n^2 e^{5u}) \exp(-\pi n^2 e^{4u}).

As discussed in this previous post, the Riemann hypothesis is equivalent to the assertion that all the zeroes of {H_0} are real.

De Bruijn and Newman showed that there existed a real constant {\Lambda} – the de Bruijn-Newman constant – such that {H_t} has all zeroes real whenever {t \geq \Lambda}, and at least one non-real zero when {t < \Lambda}. In particular, the Riemann hypothesis is equivalent to the upper bound {\Lambda \leq 0}. In the opposite direction, several lower bounds on {\Lambda} have been obtained over the years, most recently in my paper with Brad Rodgers where we showed that {\Lambda \geq 0}, a conjecture of Newman.

As for upper bounds, de Bruijn showed back in 1950 that {\Lambda \leq 1/2}. The only progress since then has been the work of Ki, Kim and Lee in 2009, who improved this slightly to {\Lambda < 1/2}. The primary proposed aim of this Polymath project is to obtain further explicit improvements to the upper bound of {\Lambda}. Of course, if we could lower the upper bound all the way to zero, this would solve the Riemann hypothesis, but I do not view this as a realistic outcome of this project; rather, the upper bounds that one could plausibly obtain by known methods and numerics would be comparable in achievement to the various numerical verifications of the Riemann hypothesis that exist in the literature (e.g., that the first {N} non-trivial zeroes of the zeta function lie on the critical line, for various large explicit values of {N}).

In addition to the primary goal, one could envisage some related secondary goals of the project, such as a better understanding (both analytic and numerical) of the functions {H_t} (or of similar functions), and of the dynamics of the zeroes of these functions. Perhaps further potential goals could emerge in the discussion to this post.

I think there is a plausible plan of attack on this project that proceeds as follows. Firstly, there are results going back to the original work of de Bruijn that demonstrate that the zeroes of {H_t} become attracted to the real line as {t} increases; in particular, if one defines {\sigma_{max}(t)} to be the supremum of the imaginary parts of all the zeroes of {H_t}, then it is known that this quantity obeys the differential inequality

\displaystyle \frac{d}{dt} \sigma_{max}(t) \leq - \frac{1}{\sigma_{max}(t)} \ \ \ \ \ (1)

 

whenever {\sigma_{max}(t)} is positive; furthermore, once {\sigma_{max}(t) = 0} for some {t}, then {\sigma_{max}(t') = 0} for all {t' > t}. I hope to explain this in a future post (it is basically due to the attraction that a zero off the real axis has to its complex conjugate). As a corollary of this inequality, we have the upper bound

\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2 \ \ \ \ \ (2)

 

for any real number {t}. For instance, because all the non-trivial zeroes of the Riemann zeta function lie in the critical strip {\{ s: 0 \leq \mathrm{Re} s \leq 1 \}}, one has {\sigma_{max}(0) \leq 1}, which when inserted into (2) gives {\Lambda \leq 1/2}. The inequality (1) also gives {\sigma_{max}(t) \leq \sqrt{1-2t}} for all {0 \leq t \leq 1/2}. If we could find some explicit {t} between {0} and {1/2} where we can improve this upper bound on {\sigma_{max}(t)} by an explicit constant, this would lead to a new upper bound on {\Lambda}.

Secondly, the work of Ki, Kim and Lee (based on an analysis of the various terms appearing in the expression for {H_t}) shows that for any positive {t}, all but finitely many of the zeroes of {H_t} are real (in contrast with the {t=0} situation, where it is still an open question as to whether the proportion of non-trivial zeroes of the zeta function on the critical line is asymptotically equal to {1}). As a key step in this analysis, Ki, Kim, and Lee show that for any {t>0} and {\varepsilon>0}, there exists a {T>0} such that all the zeroes of {H_t} with real part at least {T}, have imaginary part at most {\varepsilon}. Ki, Kim and Lee do not explicitly compute how {T} depends on {t} and {\varepsilon}, but it looks like this bound could be made effective.

If so, this suggests a possible strategy to get a new upper bound on {\Lambda}:

  • Select a good choice of parameters {t, \varepsilon > 0}.
  • By refining the Ki-Kim-Lee analysis, find an explicit {T} such that all zeroes of {H_t} with real part at least {T} have imaginary part at most {\varepsilon}.
  • By a numerical computation (e.g. using the argument principle), also verify that zeroes of {H_t} with real part between {0} and {T} have imaginary part at most {\varepsilon}.
  • Combining these facts, we obtain that {\sigma_{max}(t) \leq \varepsilon}; hopefully, one can insert this into (2) and get a new upper bound for {\Lambda}.

Of course, there may also be alternate strategies to upper bound {\Lambda}, and I would imagine this would also be a legitimate topic of discussion for this project.

One appealing thing about the above strategy for the purposes of a polymath project is that it naturally splits the project into several interacting but reasonably independent parts: an analytic part in which one tries to refine the Ki-Kim-Lee analysis (based on explicitly upper and lower bounding various terms in a certain series expansion for {H_t} – I may detail this later in a subsequent post); a numerical part in which one controls the zeroes of {H_t} in a certain finite range; and perhaps also a dynamical part where one sees if there is any way to improve the inequality (2). For instance, the numerical “team” might, over time, be able to produce zero-free regions for {H_t} with an increasingly large value of {T}, while in parallel the analytic “team” might produce increasingly smaller values of {T} beyond which they can control zeroes, and eventually the two bounds would meet up and we obtain a new bound on {\Lambda}. This factoring of the problem into smaller parts was also a feature of the successful Polymath8 project on bounded gaps between primes.

The project also resembles Polymath8 in another aspect: that there is an obvious way to numerically measure progress, by seeing how the upper bound for {\Lambda} decreases over time (and presumably there will also be another metric of progress regarding how well we can control {T} in terms of {t} and {\varepsilon}). However, in Polymath8 the final measure of progress (the upper bound {H} on gaps between primes) was a natural number, and thus could not decrease indefinitely. Here, the bound will be a real number, and there is a possibility that one may end up having an infinite descent in which progress slows down over time, with refinements to increasingly less significant digits of the bound as the project progresses. Because of this, I think it makes sense to follow recent Polymath projects and place an expiration date for the project, for instance one year after the launch date, in which we will agree to end the project and (if the project was successful enough) write up the results, unless there is consensus at that time to extend the project. (In retrospect, we should probably have imposed similar sunset dates on older Polymath projects, some of which have now been inactive for years, but that is perhaps a discussion for another time.)

Some Polymath projects have been known for a breakneck pace, making it hard for some participants to keep up. It’s hard to control these things, but I am envisaging a relatively leisurely project here, perhaps taking the full year mentioned above. It may well be that as the project matures we will largely be waiting for the results of lengthy numerical calculations to come in, for instance. Of course, as with previous projects, we would maintain some wiki pages (and possibly some other resources, such as a code repository) to keep track of progress and also to summarise what we have learned so far. For instance, as was done with some previous Polymath projects, we could begin with some “online reading seminars” where we go through some relevant piece of literature (most obviously the Ki-Kim-Lee paper, but there may be other resources that become relevant, e.g. one could imagine the literature on numerical verification of RH to be of value).

One could also imagine some incidental outcomes of this project, such as a more efficient way to numerically establish zero free regions for various analytic functions of interest; in particular, the project may well end up focusing on some other aspect of mathematics than the specific questions posed here.

Anyway, I would be interested to hear in the comments below from others who might be interested in participating, or at least observing, this project, particularly if they have suggestions regarding the scope and direction of the project, and on organisational structure (e.g. if one should start with reading seminars, or some initial numerical exploration of the functions {H_t}, etc..) One could also begin some preliminary discussion of the actual mathematics of the project itself, though (in line with the leisurely pace I was hoping for), I expect that the main burst of mathematical activity would happen later, once the project is formally launched (with wiki page resources, blog posts dedicated to specific aspects of the project, etc.).

 

In this post we assume the Riemann hypothesis and the simplicity of zeroes, thus the zeroes of {\zeta} in the critical strip take the form {\frac{1}{2} \pm i \gamma_j} for some real number ordinates {0 < \gamma_1 < \gamma_2 < \dots}. From the Riemann-von Mangoldt formula, one has the asymptotic

\displaystyle  \gamma_n = (1+o(1)) \frac{2\pi}{\log n} n

as {n \rightarrow \infty}; in particular, the spacing {\gamma_{n+1} - \gamma_n} should behave like {\frac{2\pi}{\log n}} on the average. However, it can happen that some gaps are unusually small compared to other nearby gaps. For the sake of concreteness, let us define a Lehmer pair to be a pair of adjacent ordinates {\gamma_n, \gamma_{n+1}} such that

\displaystyle  \frac{1}{(\gamma_{n+1} - \gamma_n)^2} \geq 1.3 \sum_{m \neq n,n+1} \frac{1}{(\gamma_m - \gamma_n)^2} + \frac{1}{(\gamma_m - \gamma_{n+1})^2}. \ \ \ \ \ (1)

The specific value of constant {1.3} is not particularly important here; anything larger than {\frac{5}{4}} would suffice. An example of such a pair would be the classical pair

\displaystyle  \gamma_{6709} = 7005.062866\dots

\displaystyle  \gamma_{6710} = 7005.100564\dots

discovered by Lehmer. It follows easily from the main results of Csordas, Smith, and Varga that if an infinite number of Lehmer pairs (in the above sense) existed, then the de Bruijn-Newman constant {\Lambda} is non-negative. This implication is now redundant in view of the unconditional results of this recent paper of Rodgers and myself; however, the question of whether an infinite number of Lehmer pairs exist remain open.

In this post, I sketch an argument that Brad and I came up with (as initially suggested by Odlyzko) the GUE hypothesis implies the existence of infinitely many Lehmer pairs. We argue probabilistically: pick a sufficiently large number {T}, pick {n} at random from {T \log T} to {2 T \log T} (so that the average gap size is close to {\frac{2\pi}{\log T}}), and prove that the Lehmer pair condition (1) occurs with positive probability.

Introduce the renormalised ordinates {x_n := \frac{\log T}{2\pi} \gamma_n} for {T \log T \leq n \leq 2 T \log T}, and let {\varepsilon > 0} be a small absolute constant (independent of {T}). It will then suffice to show that

\displaystyle  \frac{1}{(x_{n+1} - x_n)^2} \geq

\displaystyle 1.3 \sum_{m \in [T \log T, 2T \log T]: m \neq n,n+1} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2}

\displaystyle  + \frac{1}{6\varepsilon^2}

(say) with probability {\gg \varepsilon^4 - o(1)}, since the contribution of those {m} outside of {[T \log T, 2T \log T]} can be absorbed by the {\frac{1}{\varepsilon^2}} factor with probability {o(1)}.

As one consequence of the GUE hypothesis, we have {x_{n+1} - x_n \leq \varepsilon^2} with probability {O(\varepsilon^6)}. Thus, if {E := \{ m \in [T \log T, 2T \log T]: x_{m+1} - x_m \leq \varepsilon^2 \}}, then {E} has density {O( \varepsilon^6 )}. Applying the Hardy-Littlewood maximal inequality, we see that with probability {O(\varepsilon^6)}, we have

\displaystyle  \sup_{h \geq 1} | \# E \cap [n+h, n-h] | \leq \frac{1}{10}

which implies in particular that

\displaystyle  |x_m - x_n|, |x_{m} - x_{n+1}| \gg \varepsilon^2 |m-n|

for all {m \in [T \log T, 2 T \log T] \backslash \{ n, n+1\}}. This implies in particular that

\displaystyle  \sum_{m \in [T \log T, 2T \log T]: |m-n| \geq \varepsilon^{-3}} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2} \ll \varepsilon^{-1}

and so it will suffice to show that

\displaystyle  \frac{1}{(x_{n+1} - x_n)^2}

\displaystyle \geq 1.3 \sum_{m \in [T \log T, 2T \log T]: m \neq n,n+1; |m-n| < \varepsilon^{-3}} \frac{1}{(x_m - x_n)^2} + \frac{1}{(x_m - x_{n+1})^2} + \frac{1}{5\varepsilon^2}

(say) with probability {\gg \varepsilon^4 - o(1)}.

By the GUE hypothesis (and the fact that {\varepsilon} is independent of {T}), it suffices to show that a Dyson sine process {(x_n)_{n \in {\bf Z}}}, normalised so that {x_0} is the first positive point in the process, obeys the inequality

\displaystyle  \frac{1}{(x_{1} - x_0)^2} \geq 1.3 \sum_{|m| < \varepsilon^{-3}: m \neq 0,1} \frac{1}{(x_m - x_0)^2} + \frac{1}{(x_m - x_1)^2} \ \ \ \ \ (2)

with probability {\gg \varepsilon^4}. However, if we let {A > 0} be a moderately large constant (and assume {\varepsilon} small depending on {A}), one can show using {k}-point correlation functions for the Dyson sine process (and the fact that the Dyson kernel {K(x,y) = \sin(\pi(x-y))/\pi(x-y)} equals {1} to second order at the origin) that

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} \gg \varepsilon^4

\displaystyle  {\bf E} N_{[-\varepsilon,0]} \binom{N_{[0,\varepsilon]}}{2} \ll \varepsilon^7

\displaystyle  {\bf E} \binom{N_{[-\varepsilon,0]}}{2} N_{[0,\varepsilon]} \ll \varepsilon^7

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[\varepsilon,A^{-1}]} \ll A^{-3} \varepsilon^4

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[-A^{-1}, -\varepsilon]} \ll A^{-3} \varepsilon^4

\displaystyle  {\bf E} N_{[-\varepsilon,0]} N_{[0,\varepsilon]} N_{[-k, k]}^2 \ll k^2 \varepsilon^4 \ \ \ \ \ (3)

for any natural number {k}, where {N_{I}} denotes the number of elements of the process in {I}. For instance, the expression {{\bf E} N_{[-\varepsilon,0]} \binom{N_{[0,\varepsilon]}}{2} } can be written in terms of the three-point correlation function {\rho_3(x_1,x_2,x_3) = \mathrm{det}(K(x_i,x_j))_{1 \leq i,j \leq 3}} as

\displaystyle  \int_{-\varepsilon \leq x_1 \leq 0 \leq x_2 \leq x_3 \leq \varepsilon} \rho_3( x_1, x_2, x_3 )\ dx_1 dx_2 dx_3

which can easily be estimated to be {O(\varepsilon^7)} (since {\rho_3 = O(\varepsilon^4)} in this region), and similarly for the other estimates claimed above.

Since for natural numbers {a,b}, the quantity {ab - 2 a \binom{b}{2} - 2 b \binom{a}{2} = ab (5-2a-2b)} is only positive when {a=b=1}, we see from the first three estimates that the event {E} that {N_{[-\varepsilon,0]} = N_{[0,\varepsilon]} = 1} occurs with probability {\gg \varepsilon^4}. In particular, by Markov’s inequality we have the conditional probabilities

\displaystyle  {\bf P} ( N_{[\varepsilon,A^{-1}]} \geq 1 | E ) \ll A^{-3}

\displaystyle  {\bf P} ( N_{[-A^{-1}, -\varepsilon]} \geq 1 | E ) \ll A^{-3}

\displaystyle  {\bf P} ( N_{[-k, k]} \geq A k^{5/3} | E ) \ll A^{-4} k^{-4/3}

and thus, if {A} is large enough, and {\varepsilon} small enough, it will be true with probability {\gg \varepsilon^4} that

\displaystyle  N_{[-\varepsilon,0]}, N_{[0,\varepsilon]} = 1

and

\displaystyle  N_{[A^{-1}, \varepsilon]} = N_{[\varepsilon, A^{-1}]} = 0

and simultaneously that

\displaystyle  N_{[-k,k]} \leq A k^{5/3}

for all natural numbers {k}. This implies in particular that

\displaystyle  x_1 - x_0 \leq 2\varepsilon

and

\displaystyle  |x_m - x_0|, |x_m - x_1| \gg_A |m|^{3/5}

for all {m \neq 0,1}, which gives (2) for {\varepsilon} small enough.

Remark 1 The above argument needed the GUE hypothesis for correlations up to fourth order (in order to establish (3)). It might be possible to reduce the number of correlations needed, but I do not see how to obtain the claim just using pair correlations only.

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