de Bruijn–Newman constant
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,
- [math]\displaystyle{ H(\lambda, z):=\int_{0}^{\infty} e^{\lambda u^{2}} \Phi(u) \cos (z u) d u }[/math],
where [math]\displaystyle{ \Phi }[/math] is the super-exponentially decaying function
- [math]\displaystyle{ \Phi(u) = \sum_{n=1}^{\infty} (2\pi^2n^4e^{9u}-3\pi n^2 e^{5u} ) e^{-\pi n^2 e^{4u}} }[/math]
and Λ is the unique real number with the property that H has only real zeros if and only if λ≥Λ.
The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of H(0, z) are real, the Riemann hypothesis is equivalent to the conjecture that Λ≤0.[1] Brad Rodgers and Terence Tao proved that Λ<0 cannot be true, so Riemann's 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 ≥ 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 ≥ 0,[5] which was then proven by Brad Rodgers and Terence Tao in 2018.
Upper bounds
De Bruijn's upper bound of [math]\displaystyle{ \Lambda\le 1/2 }[/math] was not improved until 2008, when Ki, Kim and Lee proved [math]\displaystyle{ \Lambda\lt 1/2 }[/math], making the inequality strict.[6]
In December 2018, the 15th Polymath project improved the bound to [math]\displaystyle{ \Lambda\leq 0.22 }[/math].[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 [math]\displaystyle{ \Lambda\leq 0.2 }[/math].[12]
Historical 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. |
| 2011 | −1.1×10−11[16] | Saouter, Yannick; Gourdon, Xavier; Demichel, Patrick |
| 2018 | ≥0[2] | Rodgers, Brad; Tao, Terence |
| 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. https://terrytao.wordpress.com/2018/01/19/the-de-bruijn-newman-constant-is-non-negativ/. (announcement post)
- ↑ 2.0 2.1 Rodgers, Brad; Tao, Terence (2020). "The de Bruijn–Newman Constant is Non-Negative" (in en). Forum of Mathematics, Pi 8: e6. 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].
- ↑ 4.0 4.1 de Bruijn, N.G. (1950). "The Roots of Triginometric Integrals". Duke Math. J. 17 (3): 197–226. doi:10.1215/s0012-7094-50-01720-0. https://pure.tue.nl/ws/files/1769368/597490.pdf.
- ↑ 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.
- ↑ 6.0 6.1 Ki, Haseo; Kim, Young-One; Lee, Jungseob (2009), "On the de Bruijn–Newman constant", Advances in Mathematics 222 (1): 281–306, doi:10.1016/j.aim.2009.04.003, ISSN 0001-8708, http://web.yonsei.ac.kr/haseo/p23-reprint.pdf (discussion).
- ↑ 7.0 7.1 D.H.J. Polymath (20 December 2018), Effective approximation of heat flow evolution of the Riemann [math]\displaystyle{ \xi }[/math]-function, and an upper bound for the de Bruijn-Newman constant, https://github.com/km-git-acc/dbn_upper_bound/blob/master/Writeup/debruijn.pdf, retrieved 23 December 2018
- ↑ Going below [math]\displaystyle{ \Lambda\leq 0.22? }[/math], 4 May 2018, https://terrytao.wordpress.com/2018/05/04/polymath15-ninth-thread-going-below-0-22/
- ↑ Zero-free regions, http://michaelnielsen.org/polymath1/index.php?title=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), doi:10.1007/s40687-019-0193-1, Bibcode: 2019arXiv190412438P
- ↑ 12.0 12.1 Platt, Dave; Trudgian, Tim (2021). "The Riemann hypothesis is true up to 3·1012". Bulletin of the London Mathematical Society 53 (3): 792–797. doi:10.1112/blms.12460.(preprint)
- ↑ Csordas, G.; Norfolk, T. S.; Varga, R. S. (1987-09-01). "A low bound for the de Bruijn-newman constant Λ" (in en). Numerische Mathematik 52 (5): 483–497. doi:10.1007/BF01400887. ISSN 0945-3245.
- ↑ te Riele, H. J. J. (1990-12-01). "A new lower bound for the de Bruijn-Newman constant" (in en). 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" (in en). Numerical Algorithms 1 (2): 305–329. doi:10.1007/BF02142328. ISSN 1572-9265. Bibcode: 1991NuAlg...1..305C.
- ↑ 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.
External links
- Weisstein, Eric W.. "de Bruijn–Newman Constant". http://mathworld.wolfram.com/deBruijn-NewmanConstant.html.
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