iv Contents

§2.6. The Chebyshev estimates 24

§2.7. The Bertrand Postulate 29

§2.8. The distance between consecutive primes 31

§2.9. Mersenne primes 32

§2.10. Conjectures on the distribution of prime numbers 33

Problems on Chapter 2 36

Chapter 3. The Riemann Zeta Function 39

§3.1. The definition of the Riemann Zeta Function 39

§3.2. Extending the Zeta Function to the half-plane σ 0 40

§3.3. The derivative of the Riemann Zeta Function 41

§3.4. The zeros of the Zeta Function 43

§3.5. Euler’s estimate ζ(2) =

π2/6

45

Problems on Chapter 3 48

Chapter 4. Setting the Stage for the Proof of the Prime Number

Theorem 51

§4.1. Key functions related to the Prime Number Theorem 51

§4.2. A closer analysis of the functions θ(x) and ψ(x) 52

§4.3. Useful estimates 53

§4.4. The Mertens estimate 55

§4.5. The M¨ obius function 56

§4.6. The divisor function 58

Problems on Chapter 4 60

Chapter 5. The Proof of the Prime Number Theorem 63

§5.1. A theorem of D. J. Newman 63

§5.2. An application of Newman’s theorem 65

§5.3. The proof of the Prime Number Theorem 66

§5.4. A review of the proof of the Prime Number Theorem 69

§5.5. The Riemann Hypothesis and the Prime Number Theorem 70

§5.6. Useful estimates involving primes 71

§5.7. Elementary proofs of the Prime Number Theorem 72

Problems on Chapter 5 72

Chapter 6. The Global Behavior of Arithmetic Functions 75

§6.1. Dirichlet series and arithmetic functions 75

§6.2. The uniqueness of representation of a Dirichlet series 77