# 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,

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

where ${\displaystyle \Phi }$ is the super-exponentially decaying function

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

and 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 ${\displaystyle \Lambda \leq 1/2}$ was not improved until 2008, when Ki, Kim and Lee proved ${\displaystyle \Lambda <1/2}$, making the inequality strict.[6]

In December 2018, the 15th Polymath project improved the bound to ${\displaystyle \Lambda \leq 0.22}$.[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 ${\displaystyle \Lambda \leq 0.2}$.[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

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. 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 Haseo Ki and Young-One Kim and Jungseob Lee (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 ${\displaystyle \xi }$-function, and an upper bound for the de Bruijn-Newman constant (PDF) (preprint), retrieved 23 December 2018
8. ^
9. ^
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.