The Sequence of Prime Numbers

This section is concerned with three questions:

1. Are there infinitely many primes?
2. Given $a, b\in \mathbb {Z}$ , are there infinitely many primes of the form $ax+b$ ?
3. How are the primes spaced along the number line?

We first show that there are infinitely many primes, then state Dirichlet's theorem that if $\gcd(a,b)=1$ , then $ax+b$ is a prime for infinitely many values of $x$ . Finally, we discuss the Prime Number Theorem which asserts that there are asymptotically $x/\log(x)$ primes less than $x$ , and we make a connection between this asymptotic formula and the Riemann Hypothesis.