The answer to the MO question here refers to a result from Montgomery and Vaughan, stating that $$\lim\sup \frac{R(x)}{x\sqrt{\log\log(x)}}>0,$$ where $R(x)$ is the approximation error $$R(x)=\sum_{n\leq x}\phi(n)-\frac{3x^2}{\pi^2}.$$

By Mellin inversion and Cauchy's theorem, this approximation error is related to the zeros of $\zeta(s)$ by $$R(x)=\int_{\sigma-i\infty}^{\sigma+i\infty}\frac{\zeta(s-1)x^sds}{\zeta(s)s},$$ where $1<\sigma<2$.

I had initially thought it was a typo, that $x$ should be under the root, that the result is conditional and appears to be the law of the iterated logarithm.

Thus, is it a typo or, if not, why does it not suggest that $\zeta(s)$ has zeros arbitrarily close to $\sigma=1$?

I am assuming it does not suggest that because no corollaries were mentioned.

sublinear(in x) lower boudn for the error would not really be a result. So the x should not be under the root. I do not oversee the rest of your question at the moment. $\endgroup$ – user9072 Feb 6 '13 at 22:46