The Prime Number Theorem Ben Green Mathematical Institute, Oxford E-mail address: [email protected] 2000 Mathematics Subject Classification. Primary Contents 0.1. Overview 1 0.2. Notation 1 0.3. Quantities 2 Chapter 1. Basic facts about the primes 3 1.1. Euclid's proof 3 1.2. Elementary bounds 3 1.3. The prime number theorem 6 Chapter 2. Arithmetic functions 9 2.1. M¨obiusinversion. 9 Chapter 3. Introducing the Riemann ζ-function 11 3.1. Dirichlet series 11 3.2. The ζ-function 11 3.3. Looking forward 13 3.4. Meromorphic continuation to Re s > 0. 14 Chapter 4. Some Fourier analysis 15 4.1. Introduction 15 4.2. Fourier analysis of Schwarz functions on R 16 4.3. Fourier analysis on Z and R=Z 20 4.4. The Poisson summation formula 24 Chapter 5. The analytic continuation and functional equation 25 5.1. The Γ-function 25 5.2. Statement of the functional equation for ζ 28 5.3. Proof of the functional equation 28 Chapter 6. The partial fraction expansion 31 6.1. Statement of the partial fraction expansion 31 6.2. The product formula for ζ 32 6.3. The size of a holomorphic function and its zeros 33 6.4. Weierstrass product of an integral function of order one: proof 36 v vi CONTENTS Chapter 7. Mellin transforms and the explicit formula 39 7.1. Definitions and statement of the formula 39 7.2. Proof of the explicit formula { overview 41 7.3. Estimates for the Mellin transforms. 43 7.4. Bounds for ζ0/ζ at judicuously chosen points 45 Chapter 8. The prime number theorem 49 Chapter 9. The zero-free region. Error terms. 53 9.1. The classical zero-free region 53 9.2. The prime number theorem with classical error term 55 9.3. The Riemann hypothesis and its implications for primes 57 Chapter 10. *An introduction to sieve theory 59 10.1. Introduction 59 10.2. Selberg's weights 60 Appendix A. *Some smooth bump functions 65 Appendix B. Infinite products 69 Bibliography 73 0.2. NOTATION 1 0.1. Overview These notes give a proof of the prime number theory, together with background on complex analysis, the Riemann ζ-function, and Fourier analysis. My main aim is to give some approximation to the \correct" proof of the theorem which, in my opinion, follows Davenport's book [1] up to a point but emphasises the role of smooth cutoff functions throughout and has no mention of \Perron's formula". Although this demands a little more sophistication on the part of the reader, the payoff is that different types of error terms are cleanly separated, and one can see very clearly the link between the primes and the zeta zeros. A technical advantage is that one may establish the prime number theorem using only the nonvanishing of ζ on Re s = 1, rather than a zero-free region. Other technical advantages are that one may get away with relatively crude estimates in many places, so there is no need for a careful asymptotic analysis of the Γ-function, for example. There are surprisingly few accounts along these lines in the literature. The closest I know if is Kowalski's nice book [2]. However, this is a little faster-paced, wider-ranging and lighter on detail than we aim to be here, and some readers may be put off by the fact that it is in French. Davenport's book is the traditional connoisseur's choice for a course of this nature. It is an excellent book. However, as mentioned above, there are some notable differences in emphasis with our approach here. The book of Iwaniec and Kowalski [3] contains a vast amount of material and is really for the expert, but it should find space on the shelf of any person wishing to describe themselves in that way. The notes have evolved over around 14 years from various courses I have given in Cambridge and Oxford, most recently \C3.8 Analytic Number Theory" in Hilary Term 2018 at the latter institution. 0.2. Notation As with any course, a certain amount of notation will be introduced as we go along. One very important point to be made at the outset is that log x always means the natural logarithm of x. Some people consider the notation ln x, which can be found in some books, tasteless. logC x is the same as (log x)C . We will very often use the notation btc, which means the greatest integer less than or equal to t. Throughout the course we will be using asymptotic notation. This is vital in handling the many inequalities and rough estimates we will encounter. Here is a 2 CONTENTS summary of the notation we will see. We suggest the reader not worry too much about this now; we will gain plenty of practice with this notation. • A B means that there is an absolute constant C > 0 such that jAj 6 CB. In this notation, A and B will typically be variable quantities, depending on some other parameter. For example, x + 1 x for x > 1, because jx + 1j 6 2x in this range. It is important to note that the constant C may be different in different instances of the notation. • A = O(B) means the same thing. • A k;l;m B means that jAj 6 CB, but now C is allowed to depend on some other parameters k; l; m. For example, kx k x, k + l + m = Ok;l;m(1). • A B is the same as B A. • O(A) means some quantity bounded in magnitude by CA for some ab- solute constant C > 0. In particular, O(1) simply means a quantity 5x bounded by an absolute positive constant. For example, 1+x = O(1) for x > 0. • A = o(B) means that jAj 6 "B as some other parameter becomes large enough. The other parameter will usually be clear from context. For 1 example, log x = o(1) (as x ! 1). • A = ok;l;m(B) means that jAj 6 "B as some other parameter becomes large enough, but how large it needs to be may depend on the other k parameters k; l; m. For example, log x = ok(1). • A ∼ B, which we read as \A is asymptotic to B" means that A = (1 + o(1))B. 0.3. Quantities In understanding analytic number theory, it is important to develop a robust intuitive feeling for the rough size of certain quantities. For example, one should be absolutely clear about the fact that, for X large, p 10 log X 0:01 log X n e n X : CHAPTER 1 Basic facts about the primes 1.1. Euclid's proof This course is largely about the prime numbers 2, 3, 5, 7, ::: . The most basic fact about them is the following, proven by Euclid over 2000 years ago. Theorem 1.1. There are infinitely many primes. Proof. Suppose not, and that p1; : : : ; pN is a complete list of the primes. Consider the number M = p1 ··· pN + 1. It must have a prime factor q. However, M is manifestly not divisible by any of p1; : : : ; pN , and so q2 = fp1; : : : ; pN g. This is a contradiction. Of course, it is possible to ask more refined questions. Does the sequence of prime numbers grow extremely rapidly (like the powers of two 1; 2; 22; 23;::: ), or slowly like the odd numbers 1; 3; 5; 7;::: , or somewhere in between? This is the question that will occupy us in this course. 1.2. Elementary bounds If X > 1 is a real number then we write π(X) for the number of primes less than or equal to X. Rather elementary methods suffice to get the correct order of magnitude for π(X). Theorem 1.2. There are constants 0 < c1 < 1 < c2 such that, for all sufficiently large X, X X c π(X) c : 1 log X 6 6 2 log X X Remarks. In asymptotic notation, we may thus assert that log X π(X) X log X , and the upper bound is saying that π(X) = O(X= log X). The lower bound immediately implies the infinitude of primes (and with a much better bound than Euclid's proof). The upper bound implies, in particular, that the density of the primes up to X, that is to say π(X)=X, tends to zero as X ! 1. 3 4 1. BASIC FACTS ABOUT THE PRIMES Proof. We begin with the lower bound, which is slightly easier. For this, it suffices to show that Y n (1.1) p > C1 p62n for some C1 > 1, and for sufficiently large n. Indeed, we then have n Y π(2n) C1 6 p 6 (2n) ; p62n n from which we obtain π(2n) log(2n) . This implies the lower bound in Theorem 1.2 when X is an even integer; the general case follows very quickly from this. 2n To prove (1.1), we look at the prime factorisation of n , which we write as 2n Y = pvp(n): n p62n Now we claim the following three facts. p (i) If p > 2n then vp(n) 6 1; v (n) (ii) For all p, p p 6 2n. 2n 4n (iii) n > 2n+1 . For items (i) and (ii) we use the formula 1 X 2n n (1.2) v (n) = b c − 2b c: p pi pi i=1 P1 m This is a consequence of the fact that the power of p dividing m! is i=1b pi c, this m m 2 sum being composed of b p c multiples of p, b p2 c multiples of p , and so on.