The zeros of the derivative of the riemann zeta function near the critical line

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Abstract

We study the horizontal distribution of zeros of ζ'(s) which are denoted as ρ' = β' + iγ'. We assume the Riemann hypothesis which implies β' ≥ 1/2 for any nonreal zero ρ', equality being possible only at a multiple zero of ζ(s). In this paper, we prove that lim inf (β' - 1/2) log γ' ≠ 0 if and only if, for any c> 0 and S = σ + it with 0 < |σ - 1/2| < c/ log t (t > t 0(c)), we have ζ'/ζ(S)=1/S-ρ̂+O(log t), where ρ̂ = 1/2 + iγ is the zero of ζ closest to s (and to the origin, if there are two such). We also show that if lim inf [β' - 1/2) log γ' ≠0, then for any c> 0 and s = σ + it (t > t 1(c)), we have logζ(S) = O ((log t)2-2σ/log log t) uniformly for 1/2 + c/ log t ≤ σ < σ1 1.

Original languageEnglish
Article numberrnn064
JournalInternational Mathematics Research Notices
Volume2008
Issue number1
DOIs
Publication statusPublished - 2008 Dec 1

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Riemann zeta function
Multiple Zeros
Derivative
Distribution of Zeros
Riemann hypothesis
Line
Zero
Equality
Horizontal
If and only if
Imply

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "We study the horizontal distribution of zeros of ζ'(s) which are denoted as ρ' = β' + iγ'. We assume the Riemann hypothesis which implies β' ≥ 1/2 for any nonreal zero ρ', equality being possible only at a multiple zero of ζ(s). In this paper, we prove that lim inf (β' - 1/2) log γ' ≠ 0 if and only if, for any c> 0 and S = σ + it with 0 < |σ - 1/2| < c/ log t (t > t 0(c)), we have ζ'/ζ(S)=1/S-ρ̂+O(log t), where ρ̂ = 1/2 + iγ is the zero of ζ closest to s (and to the origin, if there are two such). We also show that if lim inf [β' - 1/2) log γ' ≠0, then for any c> 0 and s = σ + it (t > t 1(c)), we have logζ(S) = O ((log t)2-2σ/log log t) uniformly for 1/2 + c/ log t ≤ σ < σ1 1.",
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The zeros of the derivative of the riemann zeta function near the critical line. / Ki, Haseo.

In: International Mathematics Research Notices, Vol. 2008, No. 1, rnn064, 01.12.2008.

Research output: Contribution to journalArticle

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