Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part equal to $1/2$ (i.e, lie on the critical line). Let
$$
\kappa = \liminf_{T \rightarrow \infty} \frac{N_0(T)}{N(T)}.
$$
Conrey showed $\kappa \geq 2/5$, which has the interpretation that at least two fifths of the non-trivial zeroes of $\zeta$ lie on the critical line. Feng recently improved this to $\kappa \geq 0.4128$. In a preprint posted on March 23, 2014, Preobrazhenskii and Preobrazhenskaya claim to show that $\kappa \geq 0.47$ and claim to outline a proof that $\kappa = 1$, which has the interpretation that almost all the non-trivial zeroes of $\zeta$ lie on the critical line The preprint is at http://arxiv.org/abs/1403.5786
I have two questions.
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How many or what parts of the many known consequences of the Riemann hypothesis are corollaries of the statement $\kappa = 1$?
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Best Answer
Regarding (1), The American Institute of Mathematics survey of the Riemann Hypothesis refers to $\kappa=1$ as the "100% hypothesis", see
http://www.aimath.org/WWN/rh/articles/html/35a/
"In contrast to most of the other conjectures in this section, the 100% Hypothesis is not motivated by applications to the prime numbers. Indeed, at present there are no known consequences of this hypothesis."