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

$\begin{array}{rl}& \frac{1}{2}+i7005.06266\dots \\ & \frac{1}{2}+i7005.10056\dots \end{array}$

(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

$\frac{1}{({\gamma }_n-{\gamma }_{n+1}{)}^2}\ge C\sum _{m\notin \{n,n+1\}}(\frac{1}{({\gamma }_m-{\gamma }_n{)}^2}+\frac{1}{({\gamma }_m-{\gamma }_{n+1}{)}^2})$

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