The Prime Number Theorem as a Capstone in a Complex Analysis Course

TL;DR

This paper provides an accessible, detailed proof of the prime number theorem tailored for complex analysis courses, serving as a teaching resource for instructors to incorporate advanced number theory topics into their curriculum.

Contribution

It offers a comprehensive, beginner-friendly proof of the prime number theorem suitable for undergraduate and graduate courses in complex analysis, emphasizing pedagogical clarity.

Findings

Accessible proof of the prime number theorem

Guidance for instructors on teaching advanced topics

Highlights potential pitfalls in existing proofs

Abstract

We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for any instructor who seeks to use the prime number theorem for a series of capstone lectures, a scaffold for a series of guided exercises, or as a framework for an inquiry-based course. We require almost no knowledge of number theory, for our aim is to make a complete proof of the prime number theorem widely accessible to complex analysis instructors. In particular, we highlight the potential pitfalls and subtleties that may catch the instructor unawares when using more terse sources.