Wikipedia

de Bruijn–Newman constant

Not to be confused with Von Mangoldt function.
For other uses of "Λ", see Lambda.

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,

,

where is the super-exponentially decaying function

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 edit

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 edit

De Bruijn's upper bound of   was not improved until 2008, when Ki, Kim and Lee proved  , making the inequality strict.[6]

In December 2018, the 15th Polymath project improved the bound to  .[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  .[12]

Historical bounds edit

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 edit

  1. ^ "The De Bruijn-Newman constant is non-negative". 19 January 2018. Retrieved 2018-01-19. (announcement post)
  2. ^ 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.
  3. ^ Dobner, Alexander (2020). "A New Proof of Newman's Conjecture and a Generalization". arXiv:2005.05142 [math.NT].
  4. ^ 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.
  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.
  6. ^ 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).
  7. ^ a b D.H.J. Polymath (20 December 2018), Effective approximation of heat flow evolution of the Riemann  -function, and an upper bound for the de Bruijn-Newman constant (PDF) (preprint), retrieved 23 December 2018
  8. ^ Going below  , 4 May 2018
  9. ^ Zero-free regions
  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)
  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
  12. ^ 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)
  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.

External links edit

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