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

Haseo Ki

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

Haseo Ki

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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 $β'\geqslant1/2$ for any non-real zero $ρ'$, equality being possible only at a multiple zero of $ζ(s)$. In this paper we prove that $\liminf(β'-1/2)\logγ'\not=0$ if and only if for any $c>0$ and $s=σ+it$ with $|σ-1/2| 0$ and $s=σ+it$ ($t\geqslant10$), we have $$ \logζ(s)=O(\frac{(\log t)^{2-2σ}}{\log\log t}) $$ uniformly for $1/2+c/\log t\leqslantσ\leqslantσ_1<1$.

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Haseo Ki. 2007-01-25. The zeros of the derivative of the Riemann zeta function near the critical line. https://arxiv.org/abs/math/0701726

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