## Prime Number Theorem I

I’ve been thinking a lot about the Riemann zeta function, denoted by ${\zeta(s)}$, 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

$\displaystyle \pi(x) \sim \frac{x}{\log x} \Longleftrightarrow \lim_{x \uparrow \infty} \frac{\pi(x)}{x(\log x)^{-1}} = 1,$

where ${\pi(x)}$ is the number of primes ${p \leq x}$. 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 ${\zeta(s) \neq 0}$ for ${\sigma > 1 (s = \sigma + it)}$. Actually, ${\zeta(s) \neq 0}$ for ${\sigma \geq 1}$, but proofing the claim for ${\sigma = 1}$ is more difficult and will be done later. The argument relies on the Euler product of ${\zeta(s)}$, which is

$\displaystyle \zeta(s) = \prod_{p}(1-p^{-s})^{-1}, \indent \sigma > 1$

We know that ${\zeta(\sigma) \neq 0}$ for any ${\sigma > 1}$. Taking the natural logarithm of the Euler product (and using the continuity of ${\log}$) gives

$\displaystyle -\sum_{p}\log(1-p^{-\sigma}) = \sum_{p}\sum_{k=1}^{\infty}\frac{p^{-k\sigma}}{k},$

where we use the power series expansion of ${\log z}$ (the principal branch). Since the series on the RHS is convergent with nonnegative terms, the order of summation is irrelevant. Hence, we see that

$\displaystyle \log \zeta(s) = \sum_{n=2}^{\infty}\frac{\Lambda(n)}{\log n}n^{-s}$

where ${\Lambda(n) = \log n}$ if ${n}$ is a nontrivial prime power for some ${p}$, and ${0}$ otherwise. ${\Lambda(n)}$ is called the von Mangoldt function. But ${\sum_{n=1}^{\infty}\frac{\Lambda(n)}{\log n}n^{-s}}$ is a Dirichlet series which converges for ${\sigma > 1}$. If ${\zeta(s) = 0}$ for some ${s = s_{0}}$, then ${\left|\log \zeta(s)\right| \rightarrow \infty, s \rightarrow s_{0}}$. Since the zeroes of an analytic function are isolated, we have

$\displaystyle \lim_{s \rightarrow s_{0}}\left|\log \zeta(s)\right| \leq \lim_{s \rightarrow s_{0}} \sum_{n=2}^{\infty}\left|\frac{\Lambda(n)}{\log n}\right|n^{-\sigma} \leq \sum_{n=1}^{\infty}n^{-\sigma_{0}} < \infty,$

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 ${\zeta(s) \neq 0}$ in ${\left\{s \in \mathbb{C}: \sigma > 1\right\}}$.