De Bruijn-Newman constant

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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_{\bf R} 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) = \Xi_{4t}(2z). }[/math]
  • Note: there may be a typo in the definition of [math]\displaystyle{ \Xi_\lambda }[/math] in [KKL2009], they may instead have intended to write [math]\displaystyle{ 4\lambda (\log x)^2 + 2 it \log x }[/math] in place of [math]\displaystyle{ \frac{\lambda}{4} (\log x)^2 + \frac{it}{2} \log x }[/math] in that definition. If so, the relationship would be [math]\displaystyle{ H_t(z) = \Xi_{t/4}(z/2) }[/math] instead of [math]\displaystyle{ H_t(z) = \Xi_{4t}(2z) }[/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]).

[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.

[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]

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] (formally, at least) 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 may have to be interpreted in a principal value sense. (See for instance [CSV1994, Lemma 2.4]. This lemma assumes that [math]\displaystyle{ t \gt \Lambda }[/math], but it is likely that one can extend to other [math]\displaystyle{ t \geq 0 }[/math] as well.)

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} + t \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].


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