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- Title
The uniform distribution modulo one of certain subsequences of ordinates of zeros of the zeta function.
- Authors
ÇİÇEK, FATMA; GONEK, STEVEN M.
- 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 \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*} where the ${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with $a , 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.
- Subjects
RIEMANN hypothesis; ZETA functions; REAL numbers; MULTIPLICITY (Mathematics)
- Publication
Mathematical Proceedings of the Cambridge Philosophical Society, 2024, Vol 176, Issue 3, p593
- ISSN
0305-0041
- Publication type
Article
- DOI
10.1017/S0305004124000045