Lehmer pair

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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

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

More precisely, a Lehmer pair can be defined as having the property that their complex coordinates and obey the inequality

for a constant .[3]

Question dropshade.png 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.[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[edit]


  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. ^ a b 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