# De Bruijn–Newman constant

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The De Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function H(λz), where λ is a real parameter and z is a complex variable. 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. In brief, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0.

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.[1] 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.[2]

As of December 2018, the current best upper bound is ${\displaystyle \Lambda \leq 0.22}$,[3][4][5] achieved in the 15th Polymath project. A manuscript of the Polymath work was submitted to arXiv in late April 2019[6] , and has been accepted for publication by the journal Research In the Mathematical Sciences.[7]

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 conjectured that Λ ≥ 0, an intriguing counterpart to the Riemann hypothesis.[8] In January 2018 Brad Rodgers and Terence Tao published a paper on arXiv in which they claim that Λ ≥ 0.[9][10] Thus if Riemann hypothesis is true, then the value of Λ must be 0.

Year Lower bound on Λ
1988 −50
1991 −5
1990 −0.385
1994 −4.379×10−6
1993 −5.895×10−9[11]
2000 −2.7×10−9[12]
2011 −1.1×10−11[13]
2018 0[9][10]

Since ${\displaystyle H(\lambda ,z)}$ is just the Fourier transform of ${\displaystyle F(e^{\lambda x}\Phi )}$ then H has the Wiener–Hopf representation:

${\displaystyle \xi (1/2+iz)=A{\sqrt {\pi }}(\lambda )^{-1}\int _{-\infty }^{\infty }e^{{\frac {-1}{4\lambda }}(x-z)^{2}}H(\lambda ,x)\,dx}$

which is only valid for λ positive or 0, it can be seen that in the limit λ tends to zero then ${\displaystyle H(0,x)=\xi (1/2+ix)}$ for the case Lambda is negative then H is defined so:

${\displaystyle H(z,\lambda )=B{\sqrt {\pi }}(\lambda )^{-1}\int _{-\infty }^{\infty }e^{{\frac {-1}{4\lambda }}(x-z)^{2}}\xi (1/2+ix)\,dx}$

where A and B are real constants.

## References

1. ^ 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.
2. ^ 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).
3. ^ 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
4. ^
5. ^
6. ^ 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)
7. ^ 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
8. ^ 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.
9. ^ a b Rodgers, Brad; Tao, Terence (2018). "The De Bruijn–Newman constant is non-negative". arXiv:1801.05914 [math.NT]. (preprint)
10. ^ a b "The De Bruijn-Newman constant is non-negative". Retrieved 2018-01-19. (announcement post)
11. ^ 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.
12. ^ Odlyzko, A.M. (2000). "An improved bound for the de Bruijn–Newman constant". Numerical Algorithms. 25: 293–303. Bibcode:2000NuAlg..25..293O. doi:10.1023/A:1016677511798. Zbl 0967.11034.
13. ^ 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.