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

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

(the 6709th and 6710th zeros of the zeta function).[2]

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 ${\displaystyle \gamma _{n}}$ and ${\displaystyle \gamma _{n+1}}$ obey the inequality

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

for a constant ${\displaystyle C>5/4}$.[3]

It is an unsolved problem whether there exist infinitely many Lehmer pairs.[3] If so, it would imply that the De Bruijn–Newman constant is non-negative, a fact that has been proven unconditionally in a preprint of Brad Rodgers and Terence Tao.[4]

See also

• Montgomery's pair correlation conjecture

References

1. Csordas, George; Smith, Wayne; Varga, Richard S. (1994), "Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis", Constructive Approximation, 10 (1): 107–129, doi:10.1007/BF01205170, MR 1260363
2. Lehmer, D. H. (1956), "On the roots of the Riemann zeta-function", Acta Mathematica, 95: 291–298, doi:10.1007/BF02401102, MR 0086082
3. Tao, Terence (January 20, 2018), "Lehmer pairs and GUE", What's New
4. Rodgers, Brad; Tao, Terence (2018), The De Bruijn–Newman constant is non-negative, arXiv:1801.05914, Bibcode:2018arXiv180105914R