De Bruijn-Newman constant

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.

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.

[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.)

Wikipedia and other references

• de Bruijn-Newman constant
• Riemann xi function

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