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

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?

• jstor.org/stable/2005976?seq=1#page_scan_tab_contents – Balarka Sen Jan 25 '16 at 11:13
• I'm not sure if I understand correctly what you mean. $\pi(n)\sim n/\ln n + \sqrt n/\ln n$ is true unconditionally, because $n/\ln n + \sqrt n/\ln n\sim n/\ln n$. – Wojowu Jan 26 '16 at 20:26
• One idea would be to consider the upper bound for the error term $\pi(n)-n/\ln n$. Is that closer to what you mean? If so, this is relevant. – Wojowu Jan 26 '16 at 20:31
• @Wojowu; i think i want to use pi(x) ~ x/logx + x/(logx)^2 + o() from your link, but as stated previously the asymptotes don't work – JMP Jan 26 '16 at 21:01
• if the RH was true, the residual would be $\pi(x) - x/\ln x = \mathcal{o}(x^{1/2+\epsilon})$ for all $\epsilon > 0$. the residual would be $\pi(x) - x/\ln x = \mathcal{O}(x^{\sigma_0} / \ln x)$ if $\zeta(s)$ had a finite number of zeros at $\Re(s) = \sigma_0$ (if the RH was false) – reuns Jan 26 '16 at 22:32

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 $$\pi(x) = {\rm Li} (x) + O\left(\sqrt x \log x\right).$$
• What would this "version with $x/\log x$" actually be, though? – Wojowu Jan 26 '16 at 20:31
• The series at the end isn't convergent for any $x$. – Wojowu Jan 26 '16 at 20:33
• What is the $a$ in the $\mathcal O(xe^{-a\sqrt{\log x}})$? – user153330 Jan 26 '16 at 20:47