# Thread: A simple big-O notation proof

1. ## A simple big-O notation proof

I am writing an extended piece of work and, without wanting to assume the prime number theorem, I have to make the replacement

$\displaystyle O\big(\psi(x)\big) = O(x)$;

that is, I want to show $\displaystyle \psi(x) = O(x)$, where as usual $\displaystyle \psi(x) =\sum_{n\leq x} \Lambda(n)$. 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?

2. 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.

3. It's alright, I think I'll probably assume the prime number theorem (or at least a 'weak' version like $\displaystyle \psi(x) \asymp x$). It's not the main focus of my work and I've got quite a few pages already on proving preliminary results. But thanks very much for the reply.