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.

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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