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\}}.

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