De Bruijn–Newman constant explained

The de Bruijn–Newman constant, denoted by

Λ 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 , z): = \int ^{inf ty} _{0} e ^{λ u ^{2}} \Phi(u) \cos (z u) du ,where \Phi is the super-exponentially decaying function\Phi(u) = \sum ^{inf ty} _{n =1} ( 2 \pi ^{2n} ^{4e} ^{9u} -3 \pi n ^{2} e ^{5u} ) e ^{- \pi n ^{2} e ^{4u}} and Λ is the unique real number with the property that H has only real zeros if and only if λ \geq Λ .

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

Λ \leq 0 .^[1] Brad Rodgers and Terence Tao proved that Λ \geq 0 , 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

H has only real zeros if λ \geq 1 / 2 , and moreover, that if H has only real zeros for some λ , H also has only real zeros if λ is replaced by any larger value.^[4] Newman proved in 1976 the existence of a constant Λ for which the "if and only if" claim holds; and this then implies that Λ is unique. Newman also conjectured that Λ \geq 0 ,^[5] which was then proven by Brad Rodgers and Terence Tao in 2018.

Upper bounds

De Bruijn's upper bound of

Λ \le 1 / 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 \leq 0 . 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 \leq 0 . 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

1. Web site: The De Bruijn-Newman constant is non-negative. 19 January 2018. 2018-01-19. (announcement post)
2. 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.
3. Dobner . Alexander . 2020. A New Proof of Newman's Conjecture and a Generalization . math.NT . 2005.05142.
4. 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.
5. 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.
6. (discussion).
7. 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)
8. 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)
9. 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.
10. 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.
11. 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.
12. 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 .
13. 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 .
14. 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.