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 $\frac{1}{2}+i\gamma$ 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 $\gamma$ are arranged in increasing order, is uniformly distributed modulo one. Here $a$ and $b$ are real numbers with $a<b$, and $m_{\gamma }$ denotes the multiplicity of the zero $\frac{1}{2}+i\gamma$. The same result holds when the $\gamma$'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 $\gamma (\log T)/2\pi$ with $\gamma \in {\Gamma }_{[a,b]}$ and $0<\gamma \le T$.