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. They are named after Derrick Henry Lehmer, who discovered the pair of zeros

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

(the 6709th and 6710th zeros of the zeta function).

Unsolved problem in mathematics:

Are there infinitely many Lehmer pairs?

(more unsolved problems in mathematics)

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

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

for a constant $C>5/4$.

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