Lehmer pair

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.^[1] They are named after Derrick Henry Lehmer, who discovered the pair of zeros

${\displaystyle {\begin{aligned}&{\tfrac {1}{2}}+i\,7005.06266\dots \\[4pt]&{\tfrac {1}{2}}+i\,7005.10056\dots \end{aligned}}}$

(the 6709th and 6710th zeros of the zeta function).^[2]

More precisely, a Lehmer pair can be defined as having the property that their complex coordinates ${\displaystyle \gamma _{n}}$ and ${\displaystyle \gamma _{n+1}}$ obey the inequality

${\displaystyle {\frac {1}{(\gamma _{n}-\gamma _{n+1})^{2}}}\geq C\sum _{m\notin \{n,n+1\}}\left({\frac {1}{(\gamma _{m}-\gamma _{n})^{2}}}+{\frac {1}{(\gamma _{m}-\gamma _{n+1})^{2}}}\right)}$

for a constant ${\displaystyle C>5/4}$.^[3]

It is an unsolved problem whether there exist infinitely many Lehmer pairs.^[3] 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.^[4]

• Montgomery's pair correlation conjecture