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,
- ,
where is the
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 was not improved until 2008, when Ki, Kim and Lee proved , making the inequality strict.[6]
In December 2018, the 15th
This bound was further slightly improved in April 2020 by Platt and Trudgian to .[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. |
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 |
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)
- ^ ISSN 2050-5086.
- arXiv:2005.05142 [math.NT].
- ^ Zbl 0038.23302.
- Zbl 0342.42007.
- ^ MR 2531375 (discussion).
- ^ 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
- ^ Going below , 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)
- 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. S2CID 234355998.(preprint)
- S2CID 124008641.
- ISSN 0945-3245.
- S2CID 22606966.
- Zbl 0807.11059. Retrieved June 1, 2012.
- Zbl 0967.11034.
- MR 2813360.