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Which would be consequences on the distribution of primes if infinite number of Riemann-zeta zeros are in the critical strip but not on the critical line?

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What you can say on $\pi(x)$ depends mainly on $\sigma_0=\sup \Re(\rho)$, and if the limit is not attained then how fast does $\Re(\rho)$ converge to $\sigma_0$, if the limit is attained is it by infinitely many zeros. The main thing to know is that it is very possible that $\sigma_0=1$ and that the strongest form of the PNT is (close to) the best possible error term for $\pi(x)-Li(x)$.

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