I'm looking for a proof of the following statements :
- The Riemann hypothesis is equivalent to $\pi(x)=\text{Li}(x)+\mathcal{O}(\sqrt{x}\log x)$
- If $\Pi_0(x):=\sum_{n=1}^{+\infty}\frac{\pi_0(x^{1/n})}{n}$ where $\pi_0(x)=\lim\limits_{\varepsilon\rightarrow 0}\frac{\pi(x-\varepsilon)+\pi(x+\varepsilon)}{2}$, then we have $$ \Pi_0(x)=\text{Li}(x)-\sum_{\rho}\text{Li}(x^{\rho})-\ln 2+\int_x^{+\infty}\frac{dt}{t(t^2-1)\ln t} $$ Here $\rho$ are the zeroes of $\zeta$ in the critical strip. I already know that $$ \psi_0(x)=x-\sum_{\rho}\frac{x^{\rho}}{\rho}-\ln 2\pi-\frac{1}{2}\ln\left(1-\frac{1}{x^2}\right) $$ in case the two formulas are related.
I can't find a paper tackling the proofs of these statements, any link would be greatly appreciated !