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

2007/01/25 by Ki, Haseo
#11M06 #11M26 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0701726

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)^2-2σloglog t) uniformly for 1/2+c/log t\leqslantσ\leqslantσ₁<1.

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