Quote:
Originally Posted by R.D. Silverman
I can't give a recommendation, I did not even know about the latter
result.

I read it over last night  Granville is a great expositor (in addition to being a firstrate mathematician). It doesn't look like their method easily extends to a correction term, since they're not finding the terms directly but rather the moments and then deriving the desired conclusion from a 'magical' theorem in statistics that says that if all the moments match the normal distribution it is normal. So it appears that 1 + o(1) is all you get.
But it struck me while reading the theorem that even more important than the correction term in variance is the correction term in mean, and it is purely elementary to improve the error term in the
average order of
from O(1) to M + O(1/log x), where the error term comes both from the fraction of numbers below x which are prime and from Mertens' theorem. I know this isn't the same as the normal order, and I doubt something so sharp could be proved at that level, but this should improve the practical performance.