Apologies for the lack of posts lately; it’s been a busy semester. This post is essentially my notes for a talk I gave in my analytic number theory class.

Our goal is to obtain bounds on the distribution of prime numbers, that is, on functions of the form ${\pi(x)}$. The closely related function

$\displaystyle \psi(x) = \sum_{n \leq x} \Lambda(n)$

turns out to be amenable to study by analytic means; here ${\Lambda(n)}$ is the von Mangolt function,

$\displaystyle \Lambda(n) = \begin{cases} \log p & \text{if } n = p^m, p \ \text{prime} \\ 0 & \text{otherwise} \end{cases}.$

Bounds on ${\psi(x)}$ will imply corresponding bounds on ${\pi(x)}$ by fairly straightforward arguments. For instance, the prime number theorem is equivalent to ${\psi(x) = x + o(x)}$.

The function ${\psi(x)}$ is naturally connected to the ${\zeta}$-function in view of the formula

$\displaystyle - \frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \Lambda(n) n^{-s}.$

In other words, ${- \frac{\zeta'}{\zeta}}$ is the Dirichlet series associated to the function ${\Lambda}$. Using the theory of Mellin inversion, we can recover partial sums ${\psi(x) = \sum_{n \leq x} \Lambda(x)}$ by integration of ${-\frac{\zeta'}{\zeta}}$ along a vertical line. That is, we have

$\displaystyle \psi(x) = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} -\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds ,$

at least for ${\sigma > 1}$, in which case the integral converges. Under hypotheses on the poles of ${-\frac{\zeta'}{\zeta}}$ (equivalently, on the zeros of ${\zeta}$), we can shift the contour appropriately, and estimate the integral to derive the prime number theorem. (more…)