De Bruijn–Newman constant explained

The de Bruijn–Newman constant, denoted by

and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function
H(λ,z)

, where
λ

is a real parameter and
z

is a complex variable. More precisely,

H(λ,

	infty
z):=\int
	0

	λu²
e

\Phi(u)\cos(zu)du

,where

\Phi

is the super-exponentially decaying function

\Phi(u)=

	infty
\sum
	n=1

(2\pi²ⁿ^4e^9u-3\pin²e^5u)

	-\pin²e^4u
e

and

is the unique real number with the property that

has only real zeros if and only if

λ\geqΛ

The constant is closely connected with Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the conjecture that

Λ\leq0

.^[1] Brad Rodgers and Terence Tao proved that

Λ\geq0

, so the Riemann hypothesis is equivalent to

Λ=0

.^[2] A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner.^[3]

History

De Bruijn showed in 1950 that

has only real zeros if

λ\geq1/2

, and moreover, that if

has only real zeros for some

also has only real zeros if

is replaced by any larger value.^[4] Newman proved in 1976 the existence of a constant

for which the "if and only if" claim holds; and this then implies that

is unique. Newman also conjectured that

Λ\geq0

,^[5] which was proven forty years later, by Brad Rodgers and Terence Tao in 2018.

Upper bounds

De Bruijn's upper bound of

Λ\le1/2

was not improved until 2008, when Ki, Kim and Lee proved

Λ<1/2

, making the inequality strict.^[6]

In December 2018, the 15th Polymath project improved the bound to

Λ\leq0.22

. A manuscript of the Polymath work was submitted to arXiv in late April 2019,^[7] and was published in the journal Research In the Mathematical Sciences in August 2019.

This bound was further slightly improved in April 2020 by Platt and Trudgian to

Λ\leq0.2

.^[8]

Historical bounds

Historical lower bounds
Year	Lower bound on Λ	Authors
1987	−50^[9]	Csordas, G.; Norfolk, T. S.; Varga, R. S.
1990	−5^[10]	te Riele, H. J. J.
1991	−0.0991^[11]	Csordas, G.; Ruttan, A.; Varga, R. S.
1993	−5.895^[12]	Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S.
2000	−2.7^[13]	Odlyzko, A.M.
2011	−1.1^[14]	Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick
2018	≥0	Rodgers, Brad; Tao, Terence

Historical upper bounds
Year	Upper bound on Λ	Authors
1950	≤ 1/2	de Bruijn, N.G.
2008	< 1/2	Ki, H.; Kim, Y-O.; Lee, J.
2019	≤ 0.22	Polymath, D.H.J.
2020	≤ 0.2	Platt, D.; Trudgian, T.

Notes and References

Web site: The De Bruijn-Newman constant is non-negative. 19 January 2018. 2018-01-19. (announcement post)
Rodgers. Brad. Tao. Terence. Terence Tao. The de Bruijn–Newman Constant is Non-Negative. 2020. Forum of Mathematics, Pi. en. 8. e6. 10.1017/fmp.2020.6. 2050-5086. free. 1801.05914.
Dobner . Alexander . 2020. A New Proof of Newman's Conjecture and a Generalization . math.NT . 2005.05142.
de Bruijn. N.G.. Nicolaas Govert de Bruijn. 1950. The Roots of Triginometric Integrals. Duke Math. J.. 17. 3. 197–226. 10.1215/s0012-7094-50-01720-0. 0038.23302.
Newman. C.M.. 1976. Fourier Transforms with only Real Zeros. Proc. Amer. Math. Soc.. 61. 2. 245–251. 10.1090/s0002-9939-1976-0434982-5. 0342.42007. free.
(discussion).
1904.12438. Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant. Polymath . D.H.J. . math.NT. 2019. (preprint)
2004.09765. The Riemann hypothesis is true up to 3·1012. Platt . Dave. Trudgian . Tim. Bulletin of the London Mathematical Society. 2021. 53. 3. 792–797. 10.1112/blms.12460. 234355998. (preprint)
Csordas. G.. Norfolk. T. S.. Varga. R. S.. 1987-09-01. A low bound for the de Bruijn-newman constant Λ. Numerische Mathematik. en. 52. 5. 483–497. 10.1007/BF01400887. 124008641. 0945-3245.
te Riele. H. J. J.. 1990-12-01. A new lower bound for the de Bruijn-Newman constant. Numerische Mathematik. en. 58. 1. 661–667. 10.1007/BF01385647. 0945-3245.
Csordas. G.. Ruttan. A.. Varga. R. S.. 1991-06-01. The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis. Numerical Algorithms. en. 1. 2. 305–329. 10.1007/BF02142328. 1991NuAlg...1..305C. 22606966. 1572-9265.
Csordas . G. . Odlyzko . A.M. . Andrew Odlyzko . Smith . W. . Varga . R.S. . Richard S. Varga . A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda . . 1 . 104–111 . 1993 . June 1, 2012 . 0807.11059 .
A.M. . Odlyzko . Andrew Odlyzko . An improved bound for the de Bruijn–Newman constant . Numerical Algorithms . 25 . 293–303 . 2000 . 1 . 0967.11034 . 2000NuAlg..25..293O . 10.1023/A:1016677511798 . 5824729 .
Saouter . Yannick. Gourdon . Xavier. Demichel . Patrick. 10.1090/S0025-5718-2011-02472-5. 276. Mathematics of Computation. 2813360. 2281–2287. An improved lower bound for the de Bruijn–Newman constant. 80. 2011. free.