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

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

2007-01-25

arxiv.org/abs/math/0701726v1

Mathematics

Number Theory

19 pages

Scientific paper

We study the horizontal distribution of zeros of $\zeta'(s)$ which are denoted as $\rho'=\beta'+i\gamma'$. We assume the Riemann hypothesis which implies $\beta'\geqslant1/2$ for any non-real zero $\rho'$, equality being possible only at a multiple zero of $\zeta(s)$. In this paper we prove that $\liminf(\beta'-1/2)\log\gamma'\not=0$ if and only if for any $c>0$ and $s=\sigma+it$ with $|\sigma-1/2|0$ and $s=\sigma+it$ ($t\geqslant10$), we have $$ \log\zeta(s)=O(\frac{(\log t)^{2-2\sigma}}{\log\log t}) $$ uniformly for $1/2+c/\log t\leqslant\sigma\leqslant\sigma_1<1$.

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