### 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 < σ_{1} 1.

Original language | English |
---|---|

Article number | rnn064 |

Journal | International Mathematics Research Notices |

Volume | 2008 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2008 Dec 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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**The zeros of the derivative of the riemann zeta function near the critical line.** / Ki, Haseo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The zeros of the derivative of the riemann zeta function near the critical line

AU - Ki, Haseo

PY - 2008/12/1

Y1 - 2008/12/1

N2 - 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.

AB - 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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UR - http://www.scopus.com/inward/citedby.url?scp=77955504336&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnn064

DO - 10.1093/imrn/rnn064

M3 - Article

AN - SCOPUS:77955504336

VL - 2008

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 1

M1 - rnn064

ER -