De Bruijn-Newman constant

From Polymath Wiki
Revision as of 16:09, 12 February 2018 by Teorth (talk | contribs) (→‎Threads)
Jump to navigationJump to search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

For each real number [math]\displaystyle{ t }[/math], define the entire function [math]\displaystyle{ H_t: {\mathbf C} \to {\mathbf C} }[/math] by the formula

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

where [math]\displaystyle{ \Phi }[/math] is the super-exponentially decaying function

[math]\displaystyle{ \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}). }[/math]

It is known that [math]\displaystyle{ \Phi }[/math] is even, and that [math]\displaystyle{ H_t }[/math] is even, real on the real axis, and obeys the functional equation [math]\displaystyle{ H_t(\overline{z}) = \overline{H_t(z)} }[/math]. In particular, the zeroes of [math]\displaystyle{ H_t }[/math] are symmetric about both the real and imaginary axes. One can also express [math]\displaystyle{ H_t }[/math] in a number of different forms, such as

[math]\displaystyle{ \displaystyle H_t(z) = \frac{1}{2} \int_{\bf R} e^{tu^2} \Phi(u) e^{izu}\ du }[/math]

or

[math]\displaystyle{ \displaystyle H_t(z) = \frac{1}{2} \int_0^\infty e^{t\log^2 x} \Phi(\log x) e^{iz \log x}\ \frac{dx}{x}. }[/math]

In the notation of [KKL2009], one has

[math]\displaystyle{ \displaystyle H_t(z) = \frac{1}{8} \Xi_{t/4}(z/2). }[/math]

De Bruijn [B1950] and Newman [N1976] showed that there existed a constant, the de Bruijn-Newman constant [math]\displaystyle{ \Lambda }[/math], such that [math]\displaystyle{ H_t }[/math] has all zeroes real precisely when [math]\displaystyle{ t \geq \Lambda }[/math]. The Riemann hypothesis is equivalent to the claim that [math]\displaystyle{ \Lambda \leq 0 }[/math]. Currently it is known that [math]\displaystyle{ 0 \leq \Lambda \lt 1/2 }[/math] (lower bound in [RT2018], upper bound in [KKL2009]).

The Polymath15 project seeks to improve the upper bound on [math]\displaystyle{ \Lambda }[/math]. The current strategy is to combine the following three ingredients:

  1. Numerical zero-free regions for [math]\displaystyle{ H_t(x+iy) }[/math] of the form [math]\displaystyle{ \{ x+iy: 0 \leq x \leq T; y \geq \varepsilon \} }[/math] for explicit [math]\displaystyle{ T, \varepsilon, t \gt 0 }[/math].
  2. Rigorous asymptotics that show that [math]\displaystyle{ H_t(x+iy) }[/math] whenever [math]\displaystyle{ y \geq \varepsilon }[/math] and [math]\displaystyle{ x \geq T }[/math] for a sufficiently large [math]\displaystyle{ T }[/math].
  3. Dynamics of zeroes results that control [math]\displaystyle{ \Lambda }[/math] in terms of the maximum imaginary part of a zero of [math]\displaystyle{ H_t }[/math].

[math]\displaystyle{ t=0 }[/math]

When [math]\displaystyle{ t=0 }[/math], one has

[math]\displaystyle{ \displaystyle H_0(z) = \frac{1}{8} \xi( \frac{1}{2} + \frac{iz}{2} ) }[/math]

where

[math]\displaystyle{ \displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} \Gamma(s/2) \zeta(s) }[/math]

is the Riemann xi function. In particular, [math]\displaystyle{ z }[/math] is a zero of [math]\displaystyle{ H_0 }[/math] if and only if [math]\displaystyle{ \frac{1}{2} + \frac{iz}{2} }[/math] is a non-trivial zero of the Riemann zeta function. Thus, for instance, the Riemann hypothesis is equivalent to all the zeroes of [math]\displaystyle{ H_0 }[/math] being real, and Riemann-von Mangoldt formula (in the explicit form given by Backlund) gives

[math]\displaystyle{ \displaystyle N_0(T) - (\frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} - \frac{7}{8})| \lt 0.137 \log (T/2) + 0.443 \log\log(T/2) + 4.350 }[/math]

for any [math]\displaystyle{ T \gt 4 }[/math], where [math]\displaystyle{ N_0(T) }[/math] denotes the number of zeroes of [math]\displaystyle{ H_0 }[/math] with real part between 0 and T.

The first [math]\displaystyle{ 10^{13} }[/math] zeroes of [math]\displaystyle{ H_0 }[/math] (to the right of the origin) are real [G2004]. This numerical computation uses the Odlyzko-Schonhage algorithm. In [P2017] it was independently verified that all zeroes of [math]\displaystyle{ H_0 }[/math] between 0 and 61,220,092,000 were real.

[math]\displaystyle{ t\gt 0 }[/math]

For any [math]\displaystyle{ t\gt 0 }[/math], it is known that all but finitely many of the zeroes of [math]\displaystyle{ H_t }[/math] are real and simple [KKL2009, Theorem 1.3]. In fact, assuming the Riemann hypothesis, all of the zeroes of [math]\displaystyle{ H_t }[/math] are real and simple [CSV1994, Corollary 2].

Let [math]\displaystyle{ \sigma_{max}(t) }[/math] denote the largest imaginary part of a zero of [math]\displaystyle{ H_t }[/math], thus [math]\displaystyle{ \sigma_{max}(t)=0 }[/math] if and only if [math]\displaystyle{ t \geq \Lambda }[/math]. It is known that the quantity [math]\displaystyle{ \frac{1}{2} \sigma_{max}(t)^2 + t }[/math] is non-decreasing in time whenever [math]\displaystyle{ \sigma_{max}(t)\gt 0 }[/math] (see [KKL2009, Proposition A]. In particular we have

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

for any [math]\displaystyle{ t }[/math].

The zeroes [math]\displaystyle{ z_j(t) }[/math] of [math]\displaystyle{ H_t }[/math] obey the system of ODE

[math]\displaystyle{ \partial_t z_j(t) = - \sum_{k \neq j} \frac{2}{z_k(t) - z_j(t)} }[/math]

where the sum is interpreted in a principal value sense, and excluding those times in which [math]\displaystyle{ z_j(t) }[/math] is a repeated zero. See dynamics of zeros for more details. Writing [math]\displaystyle{ z_j(t) = x_j(t) + i y_j(t) }[/math], we can write the dynamics as

[math]\displaystyle{ \partial_t x_j = - \sum_{k \neq j} \frac{2 (x_k - x_j)}{(x_k-x_j)^2 + (y_k-y_j)^2} }[/math]
[math]\displaystyle{ \partial_t y_j = \sum_{k \neq j} \frac{2 (y_k - y_j)}{(x_k-x_j)^2 + (y_k-y_j)^2} }[/math]

where the dependence on [math]\displaystyle{ t }[/math] has been omitted for brevity.

In [KKL2009, Theorem 1.4], it is shown that for any fixed [math]\displaystyle{ t\gt 0 }[/math], the number [math]\displaystyle{ N_t(T) }[/math] of zeroes of [math]\displaystyle{ H_t }[/math] with real part between 0 and T obeys the asymptotic

[math]\displaystyle{ N_t(T) = \frac{T}{4\pi} \log \frac{T}{4\pi} - \frac{T}{4\pi} + \frac{t}{16} \log T + O(1) }[/math]

as [math]\displaystyle{ T \to \infty }[/math] (caution: the error term here is not uniform in t). Also, the zeroes behave like an arithmetic progression in the sense that

[math]\displaystyle{ z_{k+1}(t) - z_k(t) = (1+o(1)) \frac{4\pi}{\log |z_k|(t)} = (1+o(1)) \frac{4\pi}{\log k} }[/math]

as [math]\displaystyle{ k \to +\infty }[/math].

See asymptotics of H_t for asymptotics of the function [math]\displaystyle{ H_t }[/math].

Threads

Other blog posts and online discussion

Code and data

Wikipedia and other references

Bibliography