## Question concerning the arithmetic average of the Euler phi function:

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?

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Pardon me for the dumb question. I can see that bounds of this type on $\psi(n)$ may be useful for the prime number theorem. What are some uses of summing $\phi(n)$ like this? – Anweshi Jan 31 2010 at 0:12
Letting $\Phi$ denote the summatory function of $\phi$ and calculating the local average $(\Phi(x) - \Phi(x-h))/h = 6x/{\pi}^2 + O(h) + O(x\log(x)/h)$ by the asymptotic estimate, we see that $\phi$ has average order of growth $6x/{\pi}^2$. Choosing $h = \sqrt{x\log(x)}$ optimally, we obtain the estimate $6x/{\pi}^2 + O(\sqrt{x\log(x)})$ for the local average. The calculation of the average order of $\phi$ yields: (1) The chance that two large integers be coprime is $6/{\pi}^2$ and thus (2) The chance that a lattice point be visible from the origin is $6/{\pi}^2$. – engelbrekt Jan 31 2010 at 10:17
Also the asymptotic estimate for the summatory function shows that there are $3x^2/{\pi}^2 + O(x\log(x))$ distinct rational numbers in the interval $[0,1]$ having a representation as a fraction in lowest terms with denominator $\leq x$. – engelbrekt Jan 31 2010 at 10:21

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.

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