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- W2108669779 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 $$frac{zeta^{'}}{zeta}(s)=frac{1}{s-hat{P}}+O(log t),$$ where $hat{p}=1/2+igamma$ 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 $log zeta(s)=Oleft(frac{(log t)^{2-2sigma} }{log log t}right)$ uniformly for 1/2+c/log t≤ σ ≤ σ 1 <1." @default.
- W2108669779 created "2016-06-24" @default.
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- W2108669779 date "2008-01-01" @default.
- W2108669779 modified "2023-10-18" @default.
- W2108669779 title "The Zeros of the Derivative of the Riemann Zeta Function Near the Critical Line" @default.
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- W2108669779 doi "https://doi.org/10.1093/imrn/rnn064" @default.
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