Complex Variables

math.NT — Number Theory

How long can an off-line zero survive the de Bruijn-Newman heat flow? An explicit unconditional lower bound for Lambda

Contributed by Pedro Caceres

Let $\Lambda$ be the de Bruijn-Newman constant. The Riemann hypothesis (RH) is equivalent to $\Lambda\le 0$, and Rodgers and Tao proved $\Lambda\ge 0$. We ask how small $\Lambda$ can be if $\zeta$ has a zero $1/2+\delta+i\gamma$ with $\delta>0$. We use de Bruijn's strip, the zero dynamics of the Polymath project, and an explicit local zero count for the heat-flow deformations $H_\tau$ that we prove here. With these we show unconditionally that $\Lambda \ge \delta^2/(70\log\gamma\log\log\gamma)$. Equivalently, every zero $\beta+i\gamma$ of $\zeta$ with $\gamma\ge 3\cdot 10^{12}$ satisfies $|\beta-1/2|\le (70\,\Lambda\log\gamma\log\log\gamma)^{1/2}$; below that height RH has been verified. We also give the bound as an explicit formula at each height, which improves slowly with $\gamma$. The proof follows a weighted sum $\sum\psi(\mathrm{Re}\,z)(\mathrm{Im}\,z)^p$ over all zeros of $H_\tau$ in the upper half-plane, which is insensitive to collisions and to other off-line zeros. For a single pair, the imaginary part obeys an exact equation whose interaction term at $\tau=0$ is the energy slope $\partial_\sigma\log|\xi|$ of the other zeros. Under local isolation and regular spacing we prove a sharper bound, linear in $\delta$. Simulations with the zeros up to height $10^4$ agree with both bounds. We do not prove RH.