MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 deleted 1 characters in body

There is information on page 68 of Montgomery and Vaughan's book, and also on page 51 of "Introduction to analytic and probabilistic number theory" by Gérald Tenenbaum. Briefly, Montgomery has established that

$$\limsup_{x \rightarrow +\infty}\frac{R(x)}{x\sqrt{\log\log(x)}} > 0$$

and similarly with the limit inferior. So there is only modest room for improvement. Unfortunately I cannot find any reference to an upper bound conditional on RH. On page 40 Tenenbaum has a reference to page 144 of Walfisz' book on exponential sums. Walfisz uses Vinogradov's method to show that

$$R(x) = O\left(x\log^{2/3}(x)(\log\log(x))^{4/3})\right)O\left(x\log^{2/3}(x)(\log\log(x))^{4/3}\right).$$

I don't own a copy of Walfisz' book, so I have no further details.

1

There is information on page 68 of Montgomery and Vaughan's book, and also on page 51 of "Introduction to analytic and probabilistic number theory" by Gérald Tenenbaum. Briefly, Montgomery has established that

$$\limsup_{x \rightarrow +\infty}\frac{R(x)}{x\sqrt{\log\log(x)}} > 0$$

and similarly with the limit inferior. So there is only modest room for improvement. Unfortunately I cannot find any reference to an upper bound conditional on RH. On page 40 Tenenbaum has a reference to page 144 of Walfisz' book on exponential sums. Walfisz uses Vinogradov's method to show that

$$R(x) = O\left(x\log^{2/3}(x)(\log\log(x))^{4/3})\right).$$

I don't own a copy of Walfisz' book, so I have no further details.