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418:{\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)}
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116:{\displaystyle {\begin{aligned}&{\tfrac {1}{2}}+i\,7005.06266\dots \\&{\tfrac {1}{2}}+i\,7005.10056\dots \end{aligned}}}
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It is an unsolved problem whether there exist infinitely many Lehmer pairs. If so, it would imply that the
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More precisely, a Lehmer pair can be defined as having the property that their complex coordinates
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506:(1994), "Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann hypothesis",
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is non-negative, a fact that has been proven unconditionally by Brad
Rodgers and
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that are unusually close to each other. They are named after
619:(2020) , "The De Bruijn–Newman constant is non-negative",
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553:(1956), "On the roots of the Riemann zeta-function",
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126:(the 6709th and 6710th zeros of the zeta function).
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16:Pair of zeros of the Riemann zeta function
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483:Montgomery's pair correlation conjecture
140:(more unsolved problems in mathematics)
136:Are there infinitely many Lehmer pairs?
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39:, who discovered the pair of zeros
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131:Unsolved problem in mathematics
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198:{\displaystyle \gamma _{n+1}}
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165:{\displaystyle \gamma _{n}}
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509:Constructive Approximation
467:De Bruijn–Newman constant
455:{\displaystyle C>5/4}
688:Analytic number theory
599:"Lehmer pairs and GUE"
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597:(January 20, 2018),
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37:Derrick Henry Lehmer
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563:: 291–298,
471:Terence Tao
25:Lehmer pair
636:1801.05914
603:What's New
489:References
104:7005.10056
74:7005.06266
669:119140820
538:122664556
383:γ
379:−
370:γ
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331:−
322:γ
281:∉
274:∑
267:≥
239:γ
235:−
226:γ
181:γ
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107:…
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682:Category
477:See also
661:4089393
641:Bibcode
579:0086082
530:1260363
31:of the
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665:S2CID
631:arXiv
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