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The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function
Part of:
Diophantine approximation, transcendental number theory
Zeta and $L$-functions: analytic theory
Published online by Cambridge University Press: 01 March 2024
Abstract
On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros 
$1/2+i\gamma$ of the Riemann zeta function, we show that the sequence where the 
\begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*}
${\gamma }$ 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 
$1/2+i{\gamma }$. 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 
$0<{\gamma }\leq T$.
MSC classification
Primary:
11M06: $zeta (s)$ and $L(s, chi)$
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 176 , Issue 3 , May 2024 , pp. 593 - 608
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society
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