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 iii

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