Difference between revisions of "De Bruijn-Newman constant"

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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.
 
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.
  
De Bruijn and Newman 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 < 1/2</math>.
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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 < 1/2</math> (lower bound in [RT2018], upper bound in [KKL2009]).
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== <math>t=0</math> ==
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When <math>t=0</math>, one has
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:<math>\displaystyle H_0(z) = \frac{1}{8} \xi( \frac{1}{2} + \frac{iz}{2} ) </math>
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 +
where
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:<math>\displaystyle \xi(s) := \frac{s(s-1)}{2} \pi^{s/2} \Gamma(s/2) \zeta(s)</math>
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 +
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.
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== <math>t>0</math> ==
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For any <math>t>0</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]
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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)>0</math> (see [KKL2009, Proposition A].  In particular we have
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 +
:<math>\displaystyle \Lambda \leq t + \frac{1}{2} \sigma_{max}(t)^2</math>
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 +
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 > \Lambda</math>, but it is likely that one can extend to other <math>t \geq 0</math> as well.)
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== Wikipedia and other references ==
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* [https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant de Bruijn-Newman constant]
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* [https://en.wikipedia.org/wiki/Riemann_Xi_function Riemann xi function]
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== Bibliography ==
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* [B1950] N. C. de Bruijn, The roots of trigonometric integrals, Duke J. Math. 17 (1950), 197–226.
 +
* [CSV1994] G. Csordas, W. Smith, R. S. Varga, [https://link.springer.com/article/10.1007/BF01205170 Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis], Constr. Approx. 10 (1994), no. 1, 107–129.
 +
* [KKL2009] H. Ki, Y. O. Kim, and J. Lee, On the de Bruijn-Newman constant, Advances in Mathematics, 22 (2009), 281–306. [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.164.5595&rep=rep1&type=pdf Citeseer]
 +
* [N1976] C. M. Newman, Fourier transforms with only real zeroes, Proc. Amer. Math. Soc. 61 (1976), 246–251.
 +
* [RT2018] B. Rodgers, T. Tao, The de Bruijn-Newman constant is negative, preprint. [https://arxiv.org/abs/1801.05914 arXiv:1801.05914]

Revision as of 13:59, 25 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.

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.

[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.)

Wikipedia and other references

Bibliography

  • [B1950] N. C. de Bruijn, The roots of trigonometric integrals, Duke J. Math. 17 (1950), 197–226.
  • [CSV1994] G. Csordas, W. Smith, R. S. Varga, Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis, Constr. Approx. 10 (1994), no. 1, 107–129.
  • [KKL2009] H. Ki, Y. O. Kim, and J. Lee, On the de Bruijn-Newman constant, Advances in Mathematics, 22 (2009), 281–306. Citeseer
  • [N1976] C. M. Newman, Fourier transforms with only real zeroes, Proc. Amer. Math. Soc. 61 (1976), 246–251.
  • [RT2018] B. Rodgers, T. Tao, The de Bruijn-Newman constant is negative, preprint. arXiv:1801.05914