What would the Riemann Hypothesis mean for the Prime Number Theorem?

Question

The Prime Number Theorem states $\pi(n)\sim \dfrac{n}{\ln n}$.

Would there be an equally simple expression if Riemann's Hypothesis were proved true?

From Chebyshev Function, would $\pi(n)\sim \dfrac{n}{\ln n} + \sqrt n\ln n$ work?

Addendum : A relevant link : https://mathoverflow.net/questions/70713/error-term-of-the-prime-number-theorem-and-the-riemann-hypothesis

Answer 1

Yes, if the RH were proved true, then the error term for $\pi(x)$ in terms of $Li(x)$ would be optimal, namely $$ | \pi(x) - Li(x) | = O(\sqrt{x}\log{x}). $$ But since we can relate $\frac{x}{\log(x)}$ with $Li(x)$, we would also obtain a version with $\frac{x}{\log(x)}$. We have $$ {\rm Li} (x) - {x\over \log x} = O \left( {x\over \log^2 x} \right) \; . $$ Formulated differently, PNT only gives $$ \pi(x)={\rm Li} (x) + O \left(x \mathrm{e}^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty $$ for some constant $a>0$, whereas with RH we get even $$ \pi(x) = {\rm Li} (x) + O\left(\sqrt x \log x\right). $$