Math 113 Prime Number Theorem Lecture Notes (Complete And Verified Study material) (13pages) LEARNEXAMS

1 The Riemann zeta function The Prime Number Theorem asserts that the number of primes less than or equal to x is approximately equal to x log x for large values of x (here and for the rest of these notes, log denotes the natural logarithm). This quantitative statement about the distribution of primes which was conjectured by several mathematicians (including Gauss) early in the nineteenth century, and was finally proved (independently) in 1896 by Hadamard and de la Vall´ee Poussin. Their proofs used fairly elaborate analytic methods. Many other proofs and generalizations of the prime number theorem have subsequently been found. In these lecture notes, we present a relatively simple proof of the Prime Number Theorem due to D. Newman (with further simplifications by D. Zagier). Our goal is to make the proof accessible for a reader who has taken a basic course in complex analysis but who does not necessarily have any background in number theory. Throughout these notes, p will always represent a prime number. We begin with some definitions and notation. Definition 1 For x ∈ R, π(x) is the number of primes less than or equal to x. Definition 2 Let f,g : R → R. We say: • f = O(g) if there exists c ∈ R such that |f| ≤ cg • f ∼ g if limx→∞ f(x) g(x) = 1. Using this notation, the Prime Number Theorem is the following statement: Theorem 1 (Prime Number Theorem) π(x) ∼ x log x . We’ll prove a large collection of auxiliary lemmas in order to establish this result, most of which will concern certain special meromorphic functions. The most important such function for our purposes is the Riemann zeta function