This is a reference-request about a very simple statement.


The Riemann hypothesis is well-known to be equivalent to
$$(1)\ \ \ \pi(x) = \mathrm{Li}(x)+O(x^{1/2} \log x)$$
and to
$$(2)\ \ \theta(x)=x+O(x^{1/2} \log^2 x).$$
Here as usual, $\pi(x) = \sum\_{p \leq x} 1$ and $\theta(x) = \sum\_{p \leq x} \log p$.



> 
> I am looking for a reference giving the proof of the equivalence between (1) and (2).
> 
> 
> 


All the analytic number theory textbooks I have looked at gives at best a proof that
$\theta(x) \sim \pi(x) \log x$, which is clearly not enough.



Equivalence of two well-known forms of (RH): reference-request
The argument that (2) implies (1) is given as equation (13.5) in Montgomery and Vaughan's **Multiplicative Number Theory I. Classical Theory**. A similar partial summation argument (probably two lines long instead of one) will establish that (1) implies (2).


