I might be able to add something to this thread, but could you please provide a citation for a proof that you consider to be involved? If not an online proof, then one that's in some common text or reference book. Thanks.
I am writing an extended piece of work and, without wanting to assume the prime number theorem, I have to make the replacement
;
that is, I want to show , where as usual . Every proof I have seen of this sort gets a little too involved (almost proving the PNT) and I really cannot see how this is not easily shown. I mean, it is nowhere near as strong as PNT. Can anyone show a simpler way?