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Let $n_{\zeta}$ denote the number of possible real parts for the non trivial zeroes of the Riemann Zeta function. RH is equivalent to $n_{\zeta}=1$, and the symmetry arising from the functional equation implies that $n_{\zeta}$ is odd.

My question is thus: do we know an upper bound for $n_{\zeta}$? Are we at least sure it is finite?

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  • $\begingroup$ I don't think a supposed uncountability would be compatible with the fact that at least 41% of these zeroes are on the critical line. $\endgroup$ Jul 26, 2016 at 16:22
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    $\begingroup$ More simply, the set of zeros is discrete, hence countable. But I don't think it is known that the set of real parts has an upper bound which is $<1$. $\endgroup$ Jul 26, 2016 at 16:39
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    $\begingroup$ it is not known if $\zeta(s)$ has a sequence of zeros whose real part converge to $1^-$. If it does, then the number of different real parts of the non-trivial zeros between $1-\epsilon$ and $1$ has to be $\mathbb{N}$ $\endgroup$
    – reuns
    Jul 26, 2016 at 17:24

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This is for example mentioned as open in a 2010 answer by Fedor Petrov to basically a duplicate question.

It might be worth noting that nothing new has been proven in these last 6 years, or it would have been big news.

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