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{align} & \tfrac 1 2 + i\,7005.06266\dots \\[4pt] & \tfrac 1 2 + i\,7005.10056\dots \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


 * $$\frac{1}{(\gamma_n-\gamma_{n+1})^2} \ge 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.