de Bruijn–Newman constant

The de Bruijn–Newman constant, denoted by $\Lambda$ and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function $H(\lambda ,z)$ , where $\lambda$ is a real parameter and $z$ is a complex variable. More precisely,

H(\lambda ,z):=\int _{0}^{\infty }e^{\lambda u^{2}}\Phi (u)\cos(zu)\,du

,

where $\Phi$ is the super-exponentially decaying function

\Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}

and $\Lambda$ is the unique real number with the property that $H$ has only real zeros if and only if $\lambda \geq \Lambda$ .

The constant is closely connected with Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the conjecture that $\Lambda \leq 0$ .^[1] Brad Rodgers and Terence Tao proved that $\Lambda \geq 0$ , so the Riemann hypothesis is equivalent to $\Lambda =0$ .^[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 $\lambda \geq 1/2$ , and moreover, that if $H$ has only real zeros for some $\lambda$ , $H$ also has only real zeros if $\lambda$ is replaced by any larger value.^[4] Newman proved in 1976 the existence of a constant $\Lambda$ for which the "if and only if" claim holds; and this then implies that $\Lambda$ is unique. Newman also conjectured that $\Lambda \geq 0$ ,^[5] which was proven forty years later, by Brad Rodgers and Terence Tao in 2018.

Upper bounds

De Bruijn's upper bound of $\Lambda \leq 1/2$ was not improved until 2008, when Ki, Kim and Lee proved $\Lambda <1/2$ , making the inequality strict.^[6]

In December 2018, the 15th Polymath project improved the bound to $\Lambda \leq 0.22$ .^[7]^[8]^[9] A manuscript of the Polymath work was submitted to arXiv in late April 2019,^[10] and was published in the journal Research In the Mathematical Sciences in August 2019.^[11]

This bound was further slightly improved in April 2020 by Platt and Trudgian to $\Lambda \leq 0.2$ .^[12]

Historical bounds

Historical lower bounds
Year	Lower bound on Λ	Authors
1987	−50^[13]	Csordas, G.; Norfolk, T. S.; Varga, R. S.
1990	−5^[14]	te Riele, H. J. J.
1991	−0.0991^[15]	Csordas, G.; Ruttan, A.; Varga, R. S.
1993	−5.895×10⁻⁹^[16]	Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S.
2000	−2.7×10⁻⁹^[17]	Odlyzko, A.M.
2011	−1.1×10⁻¹¹^[18]	Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick
2018	≥0^[2]	Rodgers, Brad; Tao, Terence

Historical upper bounds
Year	Upper bound on Λ	Authors
1950	≤ 1/2^[4]	de Bruijn, N.G.
2008	< 1/2^[6]	Ki, H.; Kim, Y-O.; Lee, J.
2019	≤ 0.22^[7]	Polymath, D.H.J.
2020	≤ 0.2^[12]	Platt, D.; Trudgian, T.

References

^ "The De Bruijn-Newman constant is non-negative". 19 January 2018. Retrieved 2018-01-19. (announcement post)
^ ^a ^b Rodgers, Brad; Tao, Terence (2020). "The de Bruijn–Newman Constant is Non-Negative". Forum of Mathematics, Pi. 8: e6. arXiv:1801.05914. doi:10.1017/fmp.2020.6. ISSN 2050-5086.
^ Dobner, Alexander (2020). "A New Proof of Newman's Conjecture and a Generalization". arXiv:2005.05142 [math.NT].
^ ^a ^b de Bruijn, N.G. (1950). "The Roots of Triginometric Integrals" (PDF). Duke Math. J. 17 (3): 197–226. doi:10.1215/s0012-7094-50-01720-0. Zbl 0038.23302.
^ Newman, C.M. (1976). "Fourier Transforms with only Real Zeros". Proc. Amer. Math. Soc. 61 (2): 245–251. doi:10.1090/s0002-9939-1976-0434982-5. Zbl 0342.42007.
^ ^a ^b Ki, Haseo; Kim, Young-One; Lee, Jungseob (2009), "On the de Bruijn–Newman constant" (PDF), Advances in Mathematics, 222 (1): 281–306, doi:10.1016/j.aim.2009.04.003, ISSN 0001-8708, MR 2531375 (discussion).
^ ^a ^b D.H.J. Polymath (20 December 2018), Effective approximation of heat flow evolution of the Riemann $\xi$ -function, and an upper bound for the de Bruijn-Newman constant (PDF) (preprint), retrieved 23 December 2018
^ Going below $\Lambda \leq 0.22?$ , 4 May 2018
^ Zero-free regions
^ Polymath, D.H.J. (2019). "Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant". arXiv:1904.12438 [math.NT].(preprint)
^ Polymath, D.H.J. (2019), "Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant", Research in the Mathematical Sciences, 6 (3), arXiv:1904.12438, Bibcode:2019arXiv190412438P, doi:10.1007/s40687-019-0193-1, S2CID 139107960
^ ^a ^b Platt, Dave; Trudgian, Tim (2021). "The Riemann hypothesis is true up to 3·1012". Bulletin of the London Mathematical Society. 53 (3): 792–797. arXiv:2004.09765. doi:10.1112/blms.12460. S2CID 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. 52 (5): 483–497. doi:10.1007/BF01400887. ISSN 0945-3245. S2CID 124008641.
^ te Riele, H. J. J. (1990-12-01). "A new lower bound for the de Bruijn-Newman constant". Numerische Mathematik. 58 (1): 661–667. doi:10.1007/BF01385647. ISSN 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. 1 (2): 305–329. Bibcode:1991NuAlg...1..305C. doi:10.1007/BF02142328. ISSN 1572-9265. S2CID 22606966.
^ Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S. (1993). "A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda" (PDF). Electronic Transactions on Numerical Analysis. 1: 104–111. Zbl 0807.11059. Retrieved June 1, 2012.
^ Odlyzko, A.M. (2000). "An improved bound for the de Bruijn–Newman constant". Numerical Algorithms. 25 (1): 293–303. Bibcode:2000NuAlg..25..293O. doi:10.1023/A:1016677511798. S2CID 5824729. Zbl 0967.11034.
^ Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick (2011). "An improved lower bound for the de Bruijn–Newman constant". Mathematics of Computation. 80 (276): 2281–2287. doi:10.1090/S0025-5718-2011-02472-5. MR 2813360.

External links

Weisstein, Eric W. "de Bruijn–Newman Constant". MathWorld.

[tao2-1] "The De Bruijn-Newman constant is non-negative". 19 January 2018. Retrieved 2018-01-19. (announcement post)

[:0-2] Rodgers, Brad; Tao, Terence (2020). "The de Bruijn–Newman Constant is Non-Negative". Forum of Mathematics, Pi. 8: e6. arXiv:1801.05914. doi:10.1017/fmp.2020.6. ISSN 2050-5086.

[3] Dobner, Alexander (2020). "A New Proof of Newman's Conjecture and a Generalization". arXiv:2005.05142 [math.NT].

[de_Bruijn_roots-4] Bruijn, N.G. (1950). "The Roots of Triginometric Integrals" (PDF). Duke Math. J. 17 (3): 197–226. doi:10.1215/s0012-7094-50-01720-0. Zbl 0038.23302.

[5] Newman, C.M. (1976). "Fourier Transforms with only Real Zeros". Proc. Amer. Math. Soc. 61 (2): 245–251. doi:10.1090/s0002-9939-1976-0434982-5. Zbl 0342.42007.

[Ki_Kim_Lee-6] Ki, Haseo; Kim, Young-One; Lee, Jungseob (2009), "On the de Bruijn–Newman constant" (PDF), Advances in Mathematics, 222 (1): 281–306, doi:10.1016/j.aim.2009.04.003, ISSN 0001-8708, MR 2531375 (discussion).

[Polymath15-7] D.H.J. Polymath (20 December 2018), Effective approximation of heat flow evolution of the Riemann $\xi$ -function, and an upper bound for the de Bruijn-Newman constant (PDF) (preprint), retrieved 23 December 2018

[8] Going below $\Lambda \leq 0.22?$ , 4 May 2018

[9] Zero-free regions

[Polymath1-10] Polymath, D.H.J. (2019). "Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant". arXiv:1904.12438 [math.NT].(preprint)

[Polymath3-11] Polymath, D.H.J. (2019), "Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant", Research in the Mathematical Sciences, 6 (3), arXiv:1904.12438, Bibcode:2019arXiv190412438P, doi:10.1007/s40687-019-0193-1, S2CID 139107960

[Platt+Trudgian-12] Platt, Dave; Trudgian, Tim (2021). "The Riemann hypothesis is true up to 3·1012". Bulletin of the London Mathematical Society. 53 (3): 792–797. arXiv:2004.09765. doi:10.1112/blms.12460. S2CID 234355998.(preprint)

[13] Csordas, G.; Norfolk, T. S.; Varga, R. S. (1987-09-01). "A low bound for the de Bruijn-newman constant Λ". Numerische Mathematik. 52 (5): 483–497. doi:10.1007/BF01400887. ISSN 0945-3245. S2CID 124008641.

[14] te Riele, H. J. J. (1990-12-01). "A new lower bound for the de Bruijn-Newman constant". Numerische Mathematik. 58 (1): 661–667. doi:10.1007/BF01385647. ISSN 0945-3245.

[15] 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. 1 (2): 305–329. Bibcode:1991NuAlg...1..305C. doi:10.1007/BF02142328. ISSN 1572-9265. S2CID 22606966.

[16] Csordas, G.; Odlyzko, A.M.; Smith, W.; Varga, R.S. (1993). "A new Lehmer pair of zeros and a new lower bound for the De Bruijn–Newman constant Lambda" (PDF). Electronic Transactions on Numerical Analysis. 1: 104–111. Zbl 0807.11059. Retrieved June 1, 2012.

[17] Odlyzko, A.M. (2000). "An improved bound for the de Bruijn–Newman constant". Numerical Algorithms. 25 (1): 293–303. Bibcode:2000NuAlg..25..293O. doi:10.1023/A:1016677511798. S2CID 5824729. Zbl 0967.11034.

[18] Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick (2011). "An improved lower bound for the de Bruijn–Newman constant". Mathematics of Computation. 80 (276): 2281–2287. doi:10.1090/S0025-5718-2011-02472-5. MR 2813360.

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