The following is the basic steps in one proof of the prime number theorem (many steps are left out)

Now,

where:

That is, differs from only when x is a prime power, the difference being . Now, via residue integration:

where the sum is over all the non-trivial zeros of the zeta function. Dividing through by x and letting x tend to infinity:

I think it can be shown that: and therefore:

and thus

I'm not sure about the order of the sum. Can someone confirm this or explain further how this sum is bounded?