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
(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 and obey the inequality
for a constant .
|Unsolved problem in mathematics:|
Are there infinitely many Lehmer pairs?(more unsolved problems in mathematics)
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 in a preprint of Brad Rodgers and Terence Tao.
- 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
- Lehmer, D. H. (1956), "On the roots of the Riemann zeta-function", Acta Mathematica, 95: 291–298, doi:10.1007/BF02401102, MR 0086082
- Tao, Terence (January 20, 2018), "Lehmer pairs and GUE", What's New
- Rodgers, Brad; Tao, Terence (2018), The De Bruijn–Newman constant is non-negative, arXiv:1801.05914, Bibcode:2018arXiv180105914R