Analytic Number Theory

Abstract

In this chapter we draw on real and complex analysis to present four beautiful theorems. The first is P. Dirichlet’s theorem that there are infinitely many primes in any arithmetic progression

$$ a,\,a + b,\,a + 2b,\,a + 3b,\, \ldots $$

(assuming a and b are relatively prime). The second, due to J. Lambek, L. Moser, and R. Wild, gives the order of the number of primitive Pythagorean triangles with area less than n. The third is the Prime Number Theorem, first proved, independently, by J. Hadamard and C. J. de la Vallée Poussin.