I’ve been thinking a lot about the Riemann zeta function, denoted by , lately on account of an analytic number theory class I’m taking this semester with Professor Barry Mazur. We’re working up to a proof of the Prime Number Theorem (PNT), which says that
where is the number of primes . As indicated in an earlier post, I’ve been reading the delightful Complex Proofs of Real Theorems, where they give a version of a short proof of the PNT originally due to Donald Newman. The next few posts will be dedicated to an expoisition of this proof.
I thought I would begin with the, perhaps, not so obvious fact that for . Actually, for , but proofing the claim for is more difficult and will be done later. The argument relies on the Euler product of , which is
We know that for any . Taking the natural logarithm of the Euler product (and using the continuity of ) gives
where we use the power series expansion of (the principal branch). Since the series on the RHS is convergent with nonnegative terms, the order of summation is irrelevant. Hence, we see that
where if is a nontrivial prime power for some , and otherwise. is called the von Mangoldt function. But is a Dirichlet series which converges for . If for some , then . Since the zeroes of an analytic function are isolated, we have
where we use the fact that a Dirichlet series is analytic in its half plane of convergence to obtain the penultimate inequality. We conclude that in .