de Bruijn and Newman introduced a deformation parameter so that the zeros of the Riemann zeta function flow according to the (backward) heat equation. They show there is a real parameter $Lambda$ such that for times $ge Lambda$, all such zeros are on the critical line. For times $<Lambda$, there exist zeros off the critical line.

The Riemann hypothesis is equivalent to $Lambda ge 0$. Newman made the complementary conjecture $Lambda le 0$, writing "This new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so."

Lower bounds for $Lambda$ are obtained from close pairs of zeros of the zeta function, so called 'Lehmer pairs.' The current bound is $-1.14times 10^{-11}<Lambda$

This talk requires only ODEs at the lower division level of sophistication.