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

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 $${\mathrm{\Gamma }}_{[a,b]}=\{\gamma :\gamma >0and\frac{\log (\mathrm{\mid }{\zeta }^{(m_{\gamma })}(\frac{1}{2}+i\gamma )\mathrm{\mid }\mathrm{/}(\log \gamma {)}^{m_{\gamma }})}{\sqrt{\frac{1}{2}\log \log \gamma }}\in [a,b]\},$$ 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 $\gamma (\log T)\mathrm{/}2\pi$ with $\gamma \in {\mathrm{\Gamma }}_{[a,b]}$ and $0<\gamma \le T$.