Lehmer pair

Short description: Pair of zeros of the Riemann zeta function

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

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

(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 [math]\displaystyle{ \gamma_n }[/math] and [math]\displaystyle{ \gamma_{n+1} }[/math] obey the inequality

[math]\displaystyle{ \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) }[/math]

for a constant [math]\displaystyle{ C\gt 5/4 }[/math].

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