Lehmer pair explained

In the study of the Riemann hypothesis, a Lehmer pair is a pair of zeros of the Riemann zeta function that are unusually close to each other. They are named after Derrick Henry Lehmer, who discovered the pair of zeros

\begin{align} & \tfrac 1 2 + i 7005 . 06266... \\[ 4 pt] & \tfrac 1 2 + i 7005 . 10056... \end{align} (the 6709th and 6710th zeros of the zeta function).

More precisely, a Lehmer pair can be defined as having the property that their complex coordinates

\gamma _{n} and \gamma _{n+1} obey the inequality(1)/((\gamma _{n-\gamma} _{n+1} ) ^{2}) \ge C\sum _{m\notin\{n,n+1\} }\left(\frac+\frac\right)

for a constant

C> 5 / 4 .

It is an unsolved problem whether there exist infinitely many Lehmer pairs.If so, it would imply that the De Bruijn–Newman constant is non-negative,a fact that has been proven unconditionally by Brad Rodgers and Terence Tao.

See also

• Montgomery's pair correlation conjecture