Difference between revisions of "De Bruijn-Newman constant"

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:<math>\displaystyle H_t(z) = \frac{1}{2} \Xi_{4t}(2z).</math>
 
:<math>\displaystyle H_t(z) = \frac{1}{2} \Xi_{4t}(2z).</math>
  
* Note: there may be a typo in the definition of <math>\Xi_\lambda</math> in [KKL2009], they may instead have intended to write <math>4\lambda (\log x)^2 + 2 it \log x</math> in place of <math>\frac{\lambda}{4} (\log x)^2 + \frac{it}{2} \log x</math> in that definition.  If so, the relationship would be <math>H_t(z) = z) = \frac{1}{2} \Xi_{t/4}(z/2)</math> instead of <math>H_t(z) = z) = \frac{1}{2} \Xi_{4t}(2z)</math>.
+
* Note: there may be a typo in the definition of <math>\Xi_\lambda</math> in [KKL2009], they may instead have intended to write <math>4\lambda (\log x)^2 + 2 it \log x</math> in place of <math>\frac{\lambda}{4} (\log x)^2 + \frac{it}{2} \log x</math> in that definition.  If so, the relationship would be <math>H_t(z) = \frac{1}{2} \Xi_{t/4}(z/2)</math> instead of <math>H_t(z) = \frac{1}{2} \Xi_{4t}(2z)</math>.
  
  

Revision as of 10:28, 26 January 2018

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

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

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

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

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

or

[math]\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 H_t(z) = \frac{1}{2} \Xi_{4t}(2z).[/math]
  • Note: there may be a typo in the definition of [math]\Xi_\lambda[/math] in [KKL2009], they may instead have intended to write [math]4\lambda (\log x)^2 + 2 it \log x[/math] in place of [math]\frac{\lambda}{4} (\log x)^2 + \frac{it}{2} \log x[/math] in that definition. If so, the relationship would be [math]H_t(z) = \frac{1}{2} \Xi_{t/4}(z/2)[/math] instead of [math]H_t(z) = \frac{1}{2} \Xi_{4t}(2z)[/math].


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

[math]t=0[/math]

When [math]t=0[/math], one has

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

where

[math]\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]z[/math] is a zero of [math]H_0[/math] if and only if [math]\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]H_0[/math] being real, and Riemann-von Mangoldt formula (in the explicit form given by Backlund) gives

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

The first [math]10^{13}[/math] zeroes of [math]H_0[/math] (to the right of the origin) are real [G2004]. This numerical computation uses the Odlyzko-Schonhage algorithm.

[math]t\gt0[/math]

For any [math]t\gt0[/math], it is known that all but finitely many of the zeroes of [math]H_t[/math] are real and simple [KKL2009, Theorem 1.3]

Let [math]\sigma_{max}(t)[/math] denote the largest imaginary part of a zero of [math]H_t[/math], thus [math]\sigma_{max}(t)=0[/math] if and only if [math]t \geq \Lambda[/math]. It is known that the quantity [math]\frac{1}{2} \sigma_{max}(t)^2 + t[/math] is non-decreasing in time whenever [math]\sigma_{max}(t)\gt0[/math] (see [KKL2009, Proposition A]. In particular we have

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

for any [math]t[/math].

The zeroes [math]z_j(t)[/math] of [math]H_t[/math] (formally, at least) obey the system of ODE

[math]\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]t \gt \Lambda[/math], but it is likely that one can extend to other [math]t \geq 0[/math] as well.)

In [KKL2009, Theorem 1.4], it is shown that for any fixed [math]t\gt0[/math], the number [math]N_t(T)[/math] of zeroes of [math]H_t[/math] with real part between 0 and T obeys the asymptotic

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

as [math]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] 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]k \to +\infty[/math].


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