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Lehmer pair 
Summary
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53d263ba080873f5316eda56c201c715bf6d675e)
(the 6709th and 6710th zeros of the zeta function).[2]
Are there infinitely many Lehmer pairs?
More precisely, a Lehmer pair can be defined as having the property that their complex coordinates and obey the inequality
for a constant .[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 edit
References edit
- ^ 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
- ^ Lehmer, D. H. (1956), "On the roots of the Riemann zeta-function", Acta Mathematica, 95: 291–298, doi:10.1007/BF02401102, MR 0086082
- ^ a b Tao, Terence (January 20, 2018), "Lehmer pairs and GUE", What's New
- ^ 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