The zeros of the derivative of the Riemann zeta function near the critical line
| dc.creator | Ki, Haseo | |
| dc.date | 2007-01-25 | |
| dc.date.accessioned | 2026-07-07T07:43:01Z | |
| dc.date.available | 2026-07-07T07:43:01Z | |
| dc.description | 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 $|σ-1/2|<c/\log t$ $(t\geqslant10)$ $$ \frac{ζ'}ζ(s)=\frac{1}{s-ρ}+O(\log t), $$ where $ρ=1/2+iγ$ is the closest zero of $ζ(s)$ to $s$ and the origin. We also show that if $\liminf(β'-1/2)\logγ'\not=0$, then for any $c>0$ and $s=σ+it$ ($t\geqslant10$), we have $$ \logζ(s)=O(\frac{(\log t)^{2-2σ}}{\log\log t}) $$ uniformly for $1/2+c/\log t\leqslantσ\leqslantσ_1<1$. | |
| dc.description | 19 pages | |
| dc.identifier | https://arxiv.org/abs/math/0701726 | |
| dc.identifier | http://arxiv.org/abs/math/0701726 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/122661 | |
| dc.subject | Number Theory | |
| dc.subject | 11M26 ; 11M06 | |
| dc.title | The zeros of the derivative of the Riemann zeta function near the critical line | |
| dc.type | text |