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 . The closely related function

turns out to be amenable to study by analytic means; here is the von Mangolt function,

Bounds on will imply corresponding bounds on by fairly straightforward arguments. For instance, the prime number theorem is equivalent to .

The function is naturally connected to the -function in view of the formula

In other words, is the Dirichlet series associated to the function . Using the theory of Mellin inversion, we can recover partial sums by integration of along a vertical line. That is, we have

at least for , in which case the integral converges. Under hypotheses on the poles of (equivalently, on the zeros of ), we can shift the contour appropriately, and estimate the integral to derive the prime number theorem. (more…)