Find the value of the De Bruijn–Newman constant

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $u\ge0$ put \[ \Phi(u)=\sum_{n=1}^{\infty}(2\pi^2n^4e^{9u}-3\pi n^2e^{5u})\exp(-\pi n^2e^{4u}). \] For $t\in\mathbb R$ define the entire function \[ H_t(z)=\int_0^{\infty}e^{tu^2}\Phi(u)\cos(zu)\,du\qquad(z\in\mathbb C). \] There exists a real constant $\Lambda$ for which every zero of $H_t$ is real exactly when $t\ge\Lambda$. This fixes the normalization of the de Bruijn--Newman constant. The nonnegativity $\Lambda\ge0$ is proved in the cited paper; it does not determine the exact value.

Statement recorded in the supplied source.

Determine the exact value of $\Lambda$. [S2]

Short English statement

Find the precise transition parameter at which the zeros of this deformed Riemann xi function all become real.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400062 (AMR-093-0062). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Brad Rodgers and Terence Tao, The de Bruijn--Newman constant is non-negative, arXiv:1801.05914v5 (2021) Abstract and Introduction, equations (3)--(4), pp. 1--3. https://arxiv.org/pdf/1801.05914.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Find the value of the De Bruijn–Newman constant. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.