Chapter 1 Introduction to the course
This chapter provides an overview of what we will be talking about, and explains the structure of the text. It also fixes some notation that will be used throughout.
Section 1.1 What is analytic number theory?
The fundamental questions in analytic number theory, and the ones which we focus on in this course, concern the interplay between the additive and multiplicative structures on the integers. Specifically, it is quite natural to ask questions of an additive nature about constructions which are intrinsically multiplicative. In rare cases, these questions lead us to interesting algebraic structures; for instance, the fact (due to Fermat) that every prime \(p \equiv 1 \pmod{4}\) can be written uniquely as the sum of two squares leads to the study of the ring of Gaussian integers, and the fact (due to Lagrange) that every positive integer can be written as the sum of four squares ties in nicely to quaternions. However, most additive questions about multiplicative structures admit insufficiently useful algebraic structure; for instance, one cannot use algebraic techniques alone to determine which primes can be written as the sum of two cubes.
We thus turn instead to techniques from analysis; that is, we apply continuous techniques to study discrete phenomena. This tends to be most successful when proving statements “on average”; for instance, one cannot give an exact formula for the number of primes in an interval \([1,x]\text{,}\) but we can establish an asymptotic formula and give some upper bounds on the discrepancy between the exact and asymptotic formulas.
Although this methodology turns out to be unexpectedly powerful, we must remain humbled by the fact that it is comically easy to pose open and probably extremely hard questions about prime numbers, including the following old chestnuts.
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Twin primes problem.Are there infinitely many pairs of consecutive primes which differ by 2?
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Sophie Germain problem.Are there infinitely many pairs of primes \(p,q\) such that \(q = 2p-1\text{?}\)
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Goldbach problem.Is every even integer \(n=2\) equal to the sum of two primes?
While the primary goal of the course is to expose some spectacular recent progress towards the first of these problems, and indeed it is now known that there are infinitely many pairs of consecutive primes which differ by at most \(C\) for some absolute constant \(C\text{,}\) the original question seems to remain stubbornly out of reach.
Section 1.2 Basic structure of the course
In the first part of the course, we study the asymptotic distribution of prime numbers, including in arithmetic progressions, building off of Euler's insight that the divergence of the harmonic series implies the infinitude of primes. In the process, we introduce the Riemann zeta function and the Dirichlet \(L\)-functions.
In the second parts of the course, we make more careful use of the complex-analytic properties of \(L\)-functions to obtain bounds on the error term in the prime number theorem, and similarly in arithmetic progressions.
In the third part of the course, we shift gears and adopt a more combinatorial approach, related to the notion of sieves. I like to compare this change of approach to switching from cutting hair with an electric razor to a pair of scissors; this requires more control and expertise but can yield more detailed results in certain settings. We discuss the sieve of Eratosthenes, the Brun sieve, and various versions of the Selberg upper bound sieve.
In the fourth part of the course, we introduce some versions of the large sieve, which yields improved sieving results in the setting where one is willing to aggregate many error terms together. We then apply a large sieve to prove the Bombieri–Vinogradov theorem, which gives some control in aggregate of the error terms for the prime number theorem in arithmetic progressions.
In the firth part of the course, we prove a very explicit theorem of Goldston–Pintz–Yıldırım about the distribution of primes, from 2005: if \(p_n\) denotes the \(n\)-th prime, then
\begin{equation*}
\inf_n \frac{p_{n+1} - p_n}{\log p_n} = 0.
\end{equation*}
This involves a careful combination of complex-analytic estimates on divisor sums, the Selberg sieve, and the Bombieri–Vinogradov theorem.
In the sixth part of the course, we will consider some classes of “nonabelian” \(L\)-functions and see how to use analyticity properties of these \(L\)-functions (whose proofs lie far beyond the scope of the course) to prove some equidistribution results in the spirit of Dirichlet's theorem. One class of examples is the Artin \(L\)-functions, leading to the Chebotaryov density theorem: for \(L\) a number field which is Galois over \(\QQ\) with Galois group \(G\text{,}\) this theorem predicts (among other things) the density of primes \(p\) for which the prime ideal \((p)\) in \(\ZZ\) factors in a given way in the ring of integers of \(L\text{.}\) A second class of examples is the \(L\)-functions associated to elliptic curves, leading to the Sato–Tate conjecture (now a theorem): for \(E\) an elliptic curve over \(\QQ\text{,}\) this statement predicts the distribution of the number of points on the reduction of \(E\) modulo \(p\) as the prime \(p\) varies.
Section 1.3 Notations and conventions
Subsection Basics
Throughout this course, \(\NN\) denotes the set of positive integers: \(\{1,2,\dots\}\text{.}\) Whether \(\NN\) should include 0 is a matter of some controversy, but in this course it will be more convenient to omit it.
We reserve the letter \(p\) for a prime number, and a sum or product over \(p\) without further explanation means \(p\) runs over all prime numbers. If a condition is imposed, like \(p \equiv 1 \pmod{4}\text{,}\) instead take all primes obeying that condition.
Subsection Asymptotics
Suppose we are interested in limiting behavior of some functions of \(x\) as \(x\) tends to some limit. (If otherwise unspecified we will mean \(x \to \infty\text{,}\) but it should be clear from context.) We write \(f(x) \sim g(x)\) to mean that \(\lim f(x)/g(x) = 1\text{.}\) We write \(O(f(x))\) to denote any function \(g(x)\) such that \(\limsup g(x)/f(x) < \infty\) (which may vary between instances). We write \(o(f(x))\) to denote any function \(g(x)\) such that \(\limsup g(x)/f(x) = 0\) (which may again vary between instances).
Beware that sometimes we talk about limiting behavior in one variable of functions that also depend on other variables. Unless otherwise specified, you should assume the limits are not uniform in the other variables. When they are, this will be pointed out explicitly.
Subsection Miscellaneous
As is typical in analysis, in certain proofs we will encounter a number of auxiliary constants which we will not need to track individually. In such cases, we will typically use a single symbol (such as \(c\)) to refer to all of these values, even though they need not be equal, as a way to streamline notation. However, we will give fair warning before each application of this convention.
