LTCC basic course: Analytic Number Theory Andrew Granville 3 Preface Riemann's seminal 1860 memoir showed how questions on the distribution of prime numbers are more-or-less equivalent to questions on the distribution of zeros of the Riemann zeta function. This was the starting point for the beautiful theory which is at the heart of analytic number theory, which we introduce in this mini- course. Selberg showed how sieve bounds can be obtained by optimizing values over a wide class of combinatorial objects, making them a very flexible tool. We will also see how such methods can be used to help understand primes in short intervals. Key phrases: By a conjecture we mean a proposition that has not been proven, but which is favoured by some compelling evidence. An open question is a problem where the evidence is not especially convincing in either direction. The distinction between conjecture and open question is somewhat subjective and may change over time.1 We trust, though, that there will be no similar ambiguity concerning the theorems! The exercises: In order to really learn the subject the keen student should try to fully answer the exercises. We have marked several with y if they are difficult, and occasionally yy if extremely difficult. Some exercises are embedded in the text and need to be completed to fully understand the text; there are many other exercises at the end of each chapter. Textbooks: There is a bibliography at the end of these notes of worthy books. Davenport's [Da] is, for us, the greatest introduction to the key ideas of the subject. Davenport keeps the focus narrow and gives a masterful, if terse, explanation of the proof of the prime number theorem and related issues. Similarly Bombieri's [Bo] is focussed on all there is to know about the large sieve and applications, and Crandall and Pomerance [CP] on computational issues. Titchmarsh [Ti] long ago write the definitive treatise on the Riemann zeta function, and Edwards [Edw] a wonderful historical development. Narkiewicz's historical treatise [Nar] on the distribution of prime numbers is a beautiful and informative read |- the reader is asked to find more about who did what from that source. Ribenboim's [Rib] is source of joy to professional and amateur prime seekers alike. There are two recent tracts by the modern masters of the subject, [IK] and [MV], covering a wide cross-section of topics in analytic number theory; these are much broader than what we have attempted here and are recommended to someone who wishes to gain broad expertise in our subject. |||||||||||||||{ These notes have been cobbled together out of some of the course notes from two full length courses I have taught in the past, so please excuse any repetitions. 1For example, whether or not there are odd perfect numbers was an open question; however it is now known that if one exists it is > 10300, which entitles us to conjecture that there are none. Contents Chapter 1. Proofs that there are infinitely many primes, without analysis 1 1.1. Euclid and beyond 1 1.2. Various other non-analytic proofs 3 1.3. Primes in certain arithmetic progressions 4 1.4. Prime divisors of polynomials 7 1.5. A diversion: Dynamical systems proofs 9 1.6. Formulas for primes 11 1.7. Special types of primes 13 Chapter 2. Infinitely many primes, with analysis 15 2.1. First Counting Proofs 15 2.2. Euler's proof and the Riemann zeta-function 16 2.3. Upper bound on the number of primes up to x 19 2.4. An explicit lower bound on the sum of reciprocals of the primes 19 2.5. Another explicit lower bound 20 2.6. Binomial coefficients: First bounds 20 2.7. Bertrand's postulate 21 2.8. Big Oh and other notation 23 2.9. How many prime factors does a typical integer have? 25 2.10. How many primes are there up to x? 28 2.11. The Gauss-Cram`erheuristic 30 2.12. Smoothing out the prime counting function 31 2.13. An attempt to prove the prime number theorem 33 Chapter 3. The prime number theorem 35 3.1. Partial Summation 35 3.2. Chebyshev's elementary estimates 37 3.3. Multiplicative functions and Dirichlet series 38 3.4. The average value of the divisor function and Dirichlet's hyperbola method 40 3.5. The prime number theorem and the M¨obiusfunction: proof of Theorem 3.3.1 41 3.6. Selberg's formula 43 Chapter 4. Introduction to Sieve Theory 49 4.1. The sieve of Eratosthenes 49 4.2. A first combinatorial Eratosthenes 50 4.3. Integers without large prime factors: \Smooth" or \friable" numbers 52 4.4. An upper bound on the number of smooth numbers 55 4.5. Large gaps between primes 56 5 6 CONTENTS 4.6. Exercises 59 Chapter 5. Selberg's sieve applied to an arithmetic progression 61 5.1. Selberg's sieve 61 5.2. The Fundamental Lemma of Sieve Theory 63 5.3. A stronger Brun-Titchmarsh Theorem 65 5.4. Additional exercises 67 Chapter 6. Primer on analysis 69 6.1. Inequalities 69 6.2. Fourier series 70 6.3. Poisson summation 72 6.4. The order of a function 74 6.5. Complex analysis 75 6.6. Perron's formula and its variants 77 6.7. Analytic continuation 79 6.8. The Gamma function 80 Chapter 7. Riemann's plan for proving the prime number theorem 83 7.1. A method to accurately estimate the number of primes 83 7.2. Linking number theory and complex analysis 85 7.3. The functional equation 86 7.4. The zeros of the Riemann zeta-function 86 7.5. Counting primes 87 7.6. Riemann's revolutionary formula 88 Chapter 8. The fundamental properties of ζ(s) 91 8.1. Representations of ζ(s) 91 8.2. A functional equation 91 8.3. A functional equation for the Riemann zeta function 92 8.4. 8.4. Properties of ξ(s) 92 8.5. A zero-free region for ζ(s) 93 8.6. Approximations to ζ0(s)/ζ(s) 95 8.7. On the number of zeros of ζ(s) 96 Chapter 9. The proof of the Prime Number Theorem 99 9.1. The explicit formula 99 9.2. Proving the prime number theorem 101 9.3. Assuming the Riemann Hypothesis 101 9.4. Primes in short intervals, via zeta functions 102 Chapter 10. The prime number theorem for arithmetic progressions 103 10.1. Uniformity inp arithmetic progressions 104 10.2. Breaking the x-barrier 106 10.3. Improvements to the Brun-Titchmarsh Theorem 107 10.4. To be put later 107 Chapter 11. Gaps between primes, I 109 11.1. The Gauss-Cram´erheuristic predictions about primes in short intervals 109 11.2. The prime k-tuplets conjecture 110 CONTENTS 7 11.3. A quantitative prime k-tuplets conjecture 111 11.4. Densely packed primes in short intervals 113 Chapter 12. Goldston-Pintz-Yıldırım’s argument 115 12.1. The set up 115 12.2. Evaluating the sums over n 116 12.3. Evaluating the sums using Perron's formula 117 12.4. Evaluating the sums using Fourier analysis 119 12.5. Evaluating the sums using Selberg's combinatorial approach, I 120 12.6. Sums of multiplicative functions 121 12.7. Selberg's combinatorial approach, II 121 12.8. Finding a positive difference; the proof of Theorem 10.4.1 122 Chapter 13. Goldston-Pintz-Yıldırım in higher dimensional analysis 125 13.1. The set up 125 13.2. The combinatorics 126 13.3. Sums of multiplicative functions 126 13.4. The combinatorics, II 127 13.5. Finding a positive difference 128 13.6. A special case 128 13.7. Maynard's F s, and gaps between primes 128 13.8. F as a product of one dimensional functions 129 13.9. The optimal choice 130 13.10. Tao's upper bound on ρ(F ) 132 CHAPTER 1 Proofs that there are infinitely many primes, without analysis Introduction The fundamental theorem of arithmetic states that every positive integer may be factored into a product of primes in a unique way.1 Moreover any finite product of prime numbers equals some positive integer. Therefore there is a 1-to-1 corre- spondence between positive integers and finite products of primes. This means that we can understand positive integers by decomposing them into their prime factors and studying these, just as we can understand molecules by studying the atoms of which they are composed. Once one begins to determine which integers are primes and which are not, one observes that there are many of them, though as we go further out, they seem to become sparser, a decreasing proportion of the positive integers. It is also tempting to look for patterns amongst the primes: Can we find a formula that describes all of the primes? Or at least some of them? Can we prove that there are infinitely many? And, if so, can we quickly determine how many there are up to a given point? Or at least give a good estimate? How are they distributed? Regularly, or unevenly? These questions motivate this mini-course. This first, preliminary chapter investigates what is known about the infinitude of prime numbers using only imaginative elementary arguments. This is not nec- essarily part of this mini-course but the student might enjoy playing with some of the concepts here. 1.1. Euclid and beyond Ancient Greek mathematicians knew that there are infinitely many primes. Their beautiful proof by contradiction goes as follows: Suppose that there are only finitely many primes, say k of them, which we will denote by p1 = 2 < p2 = 3 < : : : < pk: What are the prime factors of p1p2 : : : pk + 1? Since this number is > 1 it must have a prime factor (by the Fundamental Theorem of Arithmetic), and this must be some pj, for some j; 1 ≤ j ≤ k (since, by assumption, p1; p2; : : : ; pk is a list of all of the primes).