De Bruijn–Newman constant

De Bruijn showed in 1950 that ${\displaystyle H}$ has only real zeros if ${\displaystyle \lambda \geq 1/2}$, and moreover, that if ${\displaystyle H}$ has only real zeros for some ${\displaystyle \lambda }$, ${\displaystyle H}$ also has only real zeros if ${\displaystyle \lambda }$ is replaced by any larger value.^[4] Newman proved in 1976 the existence of a constant ${\displaystyle \Lambda }$ for which the "if and only if" claim holds; and this then implies that ${\displaystyle \Lambda }$ is unique. Newman also conjectured that ${\displaystyle \Lambda \geq 0}$,^[5] which was proven forty years later, by Brad Rodgers and Terence Tao in 2018.

The family ${\displaystyle H_{\lambda }}$ may be viewed as a deformation of the Riemann xi function under a heat-type equation. At ${\displaystyle \lambda =0}$, the function ${\displaystyle H_{0}}$ is essentially the Riemann xi function, written as an even Fourier transform. Varying ${\displaystyle \lambda }$ multiplies the Fourier-side kernel by ${\displaystyle e^{\lambda u^{2}}}$. Differentiating under the integral sign gives

${\displaystyle {\partial H_{\lambda } \over \partial \lambda }=-{\partial ^{2}H_{\lambda } \over \partial z^{2}},}$

so that increasing ${\displaystyle \lambda }$ evolves ${\displaystyle H_{\lambda }}$ by the backward heat equation in the variable ${\displaystyle z}$.^[6][2]

In this interpretation, the de Bruijn–Newman constant is the transition time at which the deformation changes from having non-real zeros to having only real zeros. De Bruijn's theorem says that once all zeros have become real, they remain real for all later values of the heat-flow parameter. Thus ${\displaystyle \Lambda }$ measures the stability of the real-zero property under this deformation: the Riemann hypothesis is the assertion that the undeformed function ${\displaystyle H_{0}}$ already lies on the real-zero side of the transition, while Newman's conjecture asserts that it lies exactly at the boundary rather than safely inside it.^[4][2]

Newman's conjecture is the assertion that ${\displaystyle \Lambda \geq 0}$. The proof of this lower bound by Brad Rodgers and Terence Tao proceeds by contradiction. Assuming ${\displaystyle \Lambda <0}$, they analyze the motion of the zeros of ${\displaystyle H_{t}}$ under the backwards heat-flow deformation. Their analysis forces increasingly rigid control of the zeros in the range ${\displaystyle \Lambda <t\leq 0}$. In particular, they prove that it implies that the zeros of ${\displaystyle H_{0}}$ would have to be locally close to equally spaced. They then derive a contradiction with known results on the local distribution of zeros of the Riemann zeta function, such as estimates related to Montgomery's pair correlation work.^[2]

A different proof was later given by Alexander Dobner. Dobner's method avoids the zero-dynamics and zeta-zero gap estimates used by Rodgers and Tao. In the case of the Riemann xi function, the argument shows that for every ${\displaystyle t<0}$, the deformed function ${\displaystyle \xi _{t}}$ can be approximated by a Dirichlet series

${\displaystyle \zeta _{t}(s)=\sum _{n=1}^{\infty }\exp \!\left({t \over 4}\log ^{2}n\right)n^{-s},}$

whose zeros imply the existence of zeros of ${\displaystyle \xi _{t}}$ off the critical line, equivalently non-real zeros of the corresponding ${\displaystyle H_{t}}$, where the relationship is given by

${\displaystyle H_{t}(z)={1 \over 8}\,\xi _{t}\!\left({1+iz \over 2}\right).}$

Dobner's proof also gives a generalized form of Newman's conjecture for ${\displaystyle L}$-functions in the extended Selberg class.^[7]

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.^[8]

In December 2018, the 15th Polymath project improved the bound to ${\displaystyle \Lambda \leq 0.22}$.^[9][10][11] A manuscript of the Polymath work was submitted to arXiv in late April 2019,^[12] and was published in the journal Research In the Mathematical Sciences in August 2019.^[6]

This bound was further slightly improved in April 2020 by Platt and Trudgian to ${\displaystyle \Lambda \leq 0.2}$.^[13]