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Pair correlation of the zeros of the derivative of the Riemann ξ ‐function
Author(s) -
Farmer David W.,
Gonek Steven M.,
Lee Yoonbok
Publication year - 2014
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdu026
Subject(s) - riemann hypothesis , riemann zeta function , mathematics , function (biology) , riemann xi function , explicit formulae , particular values of riemann zeta function , derivative (finance) , simple (philosophy) , mathematical analysis , correlation function (quantum field theory) , pure mathematics , arithmetic zeta function , prime zeta function , statistics , philosophy , spectral density , epistemology , evolutionary biology , financial economics , economics , biology
The complex zeros of the Riemannn zeta‐function are identical to the zeros of the Riemann ξ ‐function, ξ ( s ) . Thus, if the Riemann hypothesis (RH) is true for the zeta‐function, then it is true for ξ ( s ) . Since ξ ( s ) is entire, the zeros ofξ ' ( s ) , its derivative, would then also satisfy a Riemann hypothesis. We investigate the pair correlation function of the zeros ofξ ' ( s )under the assumption that RH is true. We then deduce consequences about the size of gaps between these zeros and the proportion of these zeros that are simple.