Let $\varphi(n)$ denote Euler's phi-function. If we let
$$ \sum_{n\leq x} \varphi(n) = \frac{3}{\pi^2}x^2+R(x),$$
then it is not hard to show that $R(x)=O(x\log x)$. What is the best known bound for $R(x)$ assuming the Riemann Hypothesis?
[Math] Question concerning the arithmetic average of the Euler phi function:
analytic-number-theorynt.number-theory
Best Answer
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 GĂ©rald 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.