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…)