Preface Preface
Last modified: December 6, 2023.
This text (still under development) is a revised version of the lecture notes of a course on analytic number theory (18.785) given at MIT in the spring of 2007. This course was an introduction to analytic number theory, including the use of zeta functions, \(L\)-functions, and sieving methods to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions). The announced prerequisites for the course were undergraduate courses in elementary number theory and complex analysis. The primary references were [3] (for zeta functions and \(L\)-functions) and [13] (for sieving methods). Many proofs were left as exercises; some more challenging ones were omitted with references given instead.
The principal goal of the original course was to present the following theorem of Goldston–Pintz–Yıldırım from 2005 (see [8], [9], [10]): for \(p_n\) the \(n\)-th prime number, we have
\begin{equation*}
\liminf_{n \to \infty} \frac{p_{n+1} - p_n}{\log p_n} = 0.
\end{equation*}
In other words, there are infinitely many pairs of consecutive primes closer together than any fixed multiple of the average spacing predicted by the prime number theorem.
Starting in 2013, these ideas were refined by Zhang [29], Maynard [16], Tao's Polymath project [21], et al. to prove bounded gaps between primes:
\begin{equation*}
\liminf_{n \to \infty} (p_{n+1} - p_n) \leq C
\end{equation*}
for some absolute constant \(C\text{;}\) as of this writing the best known bound is \(C = 246\) [21]. (The value \(C=2\) would yield the twin prime conjecture, but it is not generally believed that this circle of ideas is able to establish such a strong bound.) I am currently revising the notes with the goal of including a version of the bounded gaps theorem.
These notes reflect some comments made by participants in the original course, notably Daniel Kane and Sawyer Tabony, as well as subsequent feedback. As this document is not yet in a final state, further corrections and comments are welcome.
This document is being authored using PreTeXt, for more information on which see
pretextbook.org.
