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
1 2 + i 7005.06266 … 1 2 + i 7005.10056 … {\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).
More precisely, a Lehmer pair can be defined as having the property that their complex coordinates γ n {\displaystyle \gamma _{n}} and γ n + 1 {\displaystyle \gamma _{n+1}} obey the inequality
1 ( γ n − γ n + 1 ) 2 ≥ C ∑ m ∉ { n , n + 1 } ( 1 ( γ m − γ n ) 2 + 1 ( γ m − γ n + 1 ) 2 ) {\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 C > 5 / 4 {\displaystyle 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
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