OP-12026OpenNumber theory

de Bruijn–Newman constant ( $C_{21}$ )

de Bruijn–Newman constant: determine $C_{21}$ or improve its bounds; the repository records 0 (lower) and 0.2 (0.1875*) (upper).

MSC 11M26 · Nonreal zeros of

\zeta (s)

and

L(s, \chi)

; Riemann and other hypotheses

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓

Read original source ↗ 0 Submit progress Lean 4 ↓Discussion (0)

Problem statement

Define

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

where $\Phi$ is the super-exponential decaying function

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

Newman showed in [N1976] that there exists a finite constant $C_{21}$ (the de Bruijn–Newman constant) such that the zeros of $H$ are all real precisely when $\lambda \geq C_{21}$ .

Lean 4

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Background

The Riemann hypothesis holds iff $C_{21} = 0$ , so the lower bound likely cannot be improved upon.
Wikipedia article on the de Bruijn–Newman constant

References (from the repository page)

[N1976] Newman, Charles M. "Fourier transforms with only real zeros." Proceedings of the American Mathematical Society 61.2 (1976): 245-251. Available at https://sites.math.northwestern.edu/~auffing/papers/Newman.pdf
[B1950] de Bruijn, Nicolaas G. "The roots of trigonometric integrals." (1950): 197-226. Available at https://pure.tue.nl/ws/files/1769368/597490.pdf
[KKL2009] Ki, Haseo, Young-One Kim, and Jungseob Lee. "On the de Bruijn–Newman constant." Advances in Mathematics 222.1 (2009): 281-306. Available at https://web.archive.org/web/20170809013021/http://web.yonsei.ac.kr/haseo/p23-reprint.pdf
[P2019] Polymath, D.H.J. Effective approximation of heat flow evolution of the Riemann function, and a new upper bound for the de Bruijn–Newman constant. Res Math Sci 6, 31 (2019). arXiv:1904.12438
[PT2021] Platt, Dave, and Tim Trudgian. "The Riemann hypothesis is true up to $3 ^ 10^12$ ." Bulletin of the London Mathematical Society 53.3 (2021): 792-797. arXiv:2004.09765
[CNV1987] Csordas, George, Timothy S. Norfolk, and Richard S. Varga. "A low bound for the de Bruijn-newman constant Λ." Numerische Mathematik 52.5 (1987): 483-497. Available at https://www.math.kent.edu/~varga/pub/paper_162.pdf
[RH1990] te Riele, Herman JJ. "A new lower bound for the de Bruijn-Newman constant." Numerische Mathematik 58.1 (1990): 661-667. Available at https://ir.cwi.nl/pub/10733/10733D.pdf
[CRV1991] Csordas, George, A. Ruttan, and Richard S. Varga. "The Laguerre inequalities with applications to a problem associated with the Riemann hypothesis." Numerical Algorithms 1.2 (1991): 305-329.
[COSV1993] 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 Λ. Electron. Trans. Numer. Anal, 1, 104-111. Available at https://www2.math.ethz.ch/EMIS/journals/ETNA/vol.1.1993/pp104-111.dir/pp104-111.pdf
[O2000] Odlyzko, Andrew M. "An improved bound for the de Bruijn–Newman constant." Numerical Algorithms 25.1 (2000): 293-303. Available at https://www.researchgate.net/profile/Richard-Varga-3/publication/2267908_An_improved_bound_for_the_de_Bruijn-Newman_constant/links/558405ce08ae4738295d5f01/An-improved-bound-for-the-de-Bruijn-Newman-constant.pdf
[SGD2011] Saouter, Yannick, Xavier Gourdon, and Patrick Demichel. "An improved lower bound for the de Bruijn-Newman constant." Mathematics of Computation 80.276 (2011): 2281-2287. Available at https://www.researchgate.net/profile/Yannick-Saouter/publication/220576889_An_improved_lower_bound_for_the_de_Bruijn-Newman_constant/links/0c960532c537da7e2c000000/An-improved-lower-bound-for-the-de-Bruijn-Newman-constant.pdf
[RT2020] Rodgers, Brad, and Terence Tao. "The de Bruijn–Newman constant is non-negative." Forum of Mathematics, Pi. Vol. 8. Cambridge University Press, 2020. arXiv:1801.05914
[D2020] Dobner, Alexander. "A New Proof of Newman's Conjecture and a Generalization." arXiv:2005.05142
[MI2026] Mosaic Intelligence. "A certified unconditional upper bound Λ ≤ 0.1965 for the de Bruijn–Newman constant." 2026. Paper and certificate bundle: https://doi.org/10.5281/zenodo.20724170
[MI2026b] Mosaic Intelligence. "A certified unconditional upper bound Λ ≤ 0.1875 for the de Bruijn–Newman constant." 2026. Paper and certificate bundle: https://doi.org/10.5281/zenodo.21175533

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound 0.2 (0.1875*). Values marked with an asterisk have minimal verification.

Known upper bounds

Bound	Reference	Comments
0.5		[B1950]
< 0.5	[KKL2009]
0.22	[P2019]	Polymath project
0.2	[PT2021]
0.1965	[MI2026]	certified record package (Euler-3 mollifier route); same barrier site; see PR
0.1875	[MI2026b]	certified record package (window-tiling assembly route); same barrier site; see PR

Known lower bounds

Bound	Reference	Comments
$-50$	[CNV1987]
$-5$	[RH1990]
$-0.0991$	[CRV1991]
$-5.895 \cdot 10 ^ {-9}$	[COSV1993]
$-2.7 \cdot 10 ^ {-9}$	[O2000]
$-1.1 \cdot 10 ^{-11}$	[SGD2011]
0	[RT2020]	Simplified in [D2020]

References

[1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — de Bruijn–Newman constant ($C_{21}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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Source & evidence

https://teorth.github.io/optimizationproblems/constants/21a.html

Constant page from github.com/teorth/optimizationproblems at commit 2c1968c, Apache-2.0. The maintainers review submissions for plausibility and replicability but do not certify correctness; asterisked bounds have minimal verification.

Source date: Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differ
Imported: 2026-10-10

How we use sources ↗

Record

Difficulty: HardResisted sustained effort by specialists.
Posed by: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers) and contributors
Posed: 2026
Source: Curated problem listOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOptimization Constants in Mathematics — Davis, Ivanisvili & TaoOriginal statement ↗
Citation: Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — de Bruijn–Newman constant ($C_{21}$)
Added: 5 Nov 2026
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Fields

Number theory