What error bound would an epsilon closer to the Riemann hypothesis give?

Question

• $s=1$ line gives: $$\psi(x) = x(1+o(1))$$

• classical zero free region gives: $$\psi(x) = x + O(x e^{-c \sqrt{\log x}})$$ for some positive constant $\delta$

• RH gives: $$\psi(x) = x + O(\sqrt{x}\log(x)^2)$$

I hope these are correct, https://terrytao.wordpress.com/2009/09/24/the-prime-number-theorem-in-arithmetic-progressions-and-dueling-conspiracies/

So I was wondering what error term would we get if someone proved there were no zeros with real part $< 1-\varepsilon$?

and why does Terry Tao say understanding the error term is so important?

Answer 1

Instead of the exponent $1/2$ as in the $\sqrt x$ in the error, you get the larger $x^{1-\epsilon}$