# Primes, Riemann-zeta and zeros not on the critical line, consequences?

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?

• – Jam Feb 16 at 22:56
• One example from the linked page is that RH is equivalent to a statement involving the error term of the PNT: that for all $\varepsilon$, $\pi (n) \in \operatorname{Li}(n)+\mathcal{O}(n^{1/2+\varepsilon})$. If your premise were true, this would be false. – Jam Feb 16 at 23:08

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 $$\pi(x)-Li(x)$$.