The Uniform Distribution Modulo One of Certain Subsequences of Ordinates of Zeros of the Zeta Function

Abstract

On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $\frac12+iγ$ of the Riemann zeta function, we show that the sequence \[ Γ_{[a, b]} =\Bigg\{ γ: γ>0 \quad \mbox{and} \quad \frac{ \log\big(| ζ^{(m_γ)} (\frac12+ iγ) | / (\logγ)^{m_γ}\big)}{\sqrt{\frac12\log\logγ}} \in [a, b] \Bigg\}, \] where the $γ$ are arranged in increasing order, is uniformly distributed modulo one. Here $a$ and $b$ are real numbers with $a<b$, and $m_γ$ denotes the multiplicity of the zero $\frac12+iγ$. The same result holds when the $γ$'s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers $γ(\log T)/2π$ with $γ\in Γ_{[a, b]}$ and $0<γ\leq T$.