Contents

Preface ix

Notation xiii

Frequently Used Functions xvii

Chapter 1. Preliminary Notions 1

§1.1. Approximating a sum by an integral 1

§1.2. The Euler-MacLaurin formula 2

§1.3. The Abel summation formula 5

§1.4. Stieltjes integrals 7

§1.5. Slowly oscillating functions 8

§1.6. Combinatorial results 9

§1.7. The Chinese Remainder Theorem 10

§1.8. The density of a set of integers 11

§1.9. The Stirling formula 11

§1.10. Basic inequalities 13

Problems on Chapter 1 15

Chapter 2. Prime Numbers and Their Properties 19

§2.1. Prime numbers and their polynomial representations 19

§2.2. There exist infinitely many primes 21

§2.3. A first glimpse at the size of π(x) 21

§2.4. Fermat numbers 22

§2.5. A better lower bound for π(x) 24

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