The distribution of prime numbers has been the object of intense study by many modern mathematicians. Gauss and Legendre conjectured the prime number theorem which states that the number of primes less than a positive number \(x\) is asymptotic to \(x/\log x\) as \(x\) approaches infinity. This conjecture was later proved by Hadamard and Poisson. Their proof and many other proofs lead to what is known as Analytic Number theory. In this chapter we demonstrate elementary theorems on primes and prove elementary properties and results that will lead to the proof of the prime number theorem.
- 7.1: Introduct to Analytic Number Theory
- In this section, we show that the sum over the primes diverges as well. We also show that an interesting product will also diverge. From the following theorem, we can actually deduce that there are infinitely many primes.
- 7.2: Chebyshev's Functions
- We introduce some number theoretic functions which play important role in the distribution of primes. We also prove analytic results related to those functions.