Using Riemann-Stieltjes integration the following expression is true
$$\theta(n) = \log (t) \pi(t) \Big|_2^n - \int_2^n \frac{\pi(t)}{t}dt. \tag{1}$$
using $Li(x)$ leads to the approximation $\theta(n) \approx n $.
1) So I'm wondering using similar methods (probably based on prime number theory $\pi(x)$) if one can approximates the Chebychev $\psi(n)$ function?
2) If not is it possible and how?