The de Bruijn–Newman constant

Let the Riemann $\xi$ function evolve under the heat equation: from some time on, all its zeros are real. That threshold is the de Bruijn–Newman constant $\Lambda$. The Riemann hypothesis says exactly that $\Lambda = 0$; it is known that $0 \le \Lambda \le 0.2$, and newer computer-certified claims below 0.2 await independent checking.

Problem

Let

$$\Phi(u) = \sum_{n=1}^{\infty} \left(2\pi^2 n^4 e^{9u} - 3\pi n^2 e^{5u}\right) e^{-\pi n^2 e^{4u}}, \qquad H_t(z) = \int_0^{\infty} e^{t u^2}\, \Phi(u) \cos(z u)\, du.$$

$H_0$ is, up to scaling and a change of variable, the Riemann ξ function, and $H_t$ is essentially its evolution under the heat flow. Newman (1976) proved that there is a finite constant $\Lambda$ such that all zeros of $H_t$ are real exactly when $t \ge \Lambda$. Determine $\Lambda$, or narrow the bounds $0 \le \Lambda \le 0.2$.

Background

The Riemann hypothesis is equivalent to $\Lambda \le 0$. Newman conjectured the complementary bound $\Lambda \ge 0$, and Rodgers and Tao proved it (2020). So the Riemann hypothesis is now equivalent to $\Lambda = 0$, and each upper bound measures how close known results come to it. De Bruijn (1950) proved $\Lambda \le 1/2$; Ki, Kim and Lee (2009) made the inequality strict; the Polymath project (2019) reached $0.22$ by combining a careful analysis of the heat flow with numerical verification, and the rigorous verification of the Riemann hypothesis up to height $3 \cdot 10^{12}$ by Platt and Trudgian brought it to $0.2$.

What is known

• Lower bound: $\Lambda \ge 0$ (Rodgers and Tao, 2020), after a long series of negative lower bounds obtained from close pairs of zeta zeros, ending at $-1.1 \cdot 10^{-11}$ (Saouter, Gourdon and Demichel, 2011).
• Upper bounds, refereed: $1/2$ (de Bruijn, 1950); strictly below $1/2$ (Ki, Kim and Lee, 2009); $0.22$ (Polymath, 2019); $0.2$ (Platt and Trudgian, 2021).
• Unverified claims (asterisked on the source, all from 2026): certificate packages claim $0.1965$ and $0.1875$ (Mosaic Intelligence), $0.172422$ (Kornaus), and then $0.16$ and $0.1595$ (Mosaic Intelligence). They are self-archived and not yet independently replayed, and several of them build on the $0.1875$ package.

What would count as progress

• An independent replay of one of the certificate packages, which would turn an unverified bound into an accepted one. Each package comes with certified computations and a bundle intended for replay.
• A new upper bound below $0.2$ (or below the claimed values), with a certificate whose computations are reproducible.
• Theoretical progress that reduces the computation needed for each further step, for example sharper estimates for $H_t$ away from the region checked numerically.

References

1. C. M. Newman, "Fourier transforms with only real zeros", Proceedings of the AMS 61 (1976), 245–251. doi:10.1090/s0002-9939-1976-0434982-5
2. H. Ki, Y.-O. Kim and J. Lee, "On the de Bruijn–Newman constant", Advances in Mathematics 222 (2009), 281–306. doi:10.1016/j.aim.2009.04.003
3. B. Rodgers and T. Tao, "The de Bruijn–Newman constant is non-negative", Forum of Mathematics, Pi 8 (2020). arXiv:1801.05914
4. D. H. J. Polymath, "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 (2019), 31. arXiv:1904.12438
5. D. Platt and T. Trudgian, "The Riemann hypothesis is true up to 3·10¹²", Bulletin of the London Mathematical Society 53 (2021), 792–797. doi:10.1112/blms.12460

Source

Imported from Optimization Constants in Mathematics, constant 21, curated by Terence Tao and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.