It is well known that the Riemann Hypothesis implies the following:


$|\theta(x) - x| = O(x^{1/2 + \epsilon})$ for all $\epsilon > 0$.


where $\theta$ is the first Chebyshev function; that is, $\theta(x)$ is the sum of the logs of all the primes up to $x$. Is it known whether the reverse implication holds? If so, could you please supply the reference? Thank you.



Question about $\theta$ and the Riemann Hypthesis - reference request
More is true. Let $c\in [1/2,1)$ be a constant. It is well-known that the Riemann zeta function $\zeta(s)$ has no zeros in the half plane $\Re(s)> c$ if and only if $|\theta(x)-x|=O(x^{c+\epsilon})$ for all $\epsilon>0$.


You can find this proved in most texts on the Riemann zeta function, such as Titchmarsh's book.


