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Lehmer pair

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Pair of zeros of the Riemann zeta function

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

1 2 + i 7005.06266 … 1 2 + i 7005.10056 … {\displaystyle {\begin{aligned}&{\tfrac {1}{2}}+i\,7005.06266\dots \\&{\tfrac {1}{2}}+i\,7005.10056\dots \end{aligned}}}

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

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

1 ( γ n − γ n + 1 ) 2 ≥ C ∑ m ∉ { n , n + 1 } ( 1 ( γ m − γ n ) 2 + 1 ( γ m − γ n + 1 ) 2 ) {\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 C > 5 / 4 {\displaystyle C>5/4} .

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 by Brad Rodgers and Terence Tao.

See also

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) , "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
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