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 by 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, S2CID   122664556
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 (2020) [2018], "The De Bruijn–Newman constant is non-negative", Forum Math. Pi , 8, arXiv: 1801.05914 , Bibcode:2018arXiv180105914R, doi:10.1017/fmp.2020.6, MR   4089393, S2CID   119140820