The Series Of Primes (3)

Abstract

Abstract The functions ϑ(x)and Ψ(x). In this chapter we return to the problems concerning the distribution of primes of which we gave a preliminary account in the first two chapters. There we proved nothing except Euclid’s Theorem 4 and the slight extensions contained in §§ 2.1–6. Here we develop the theory much further and, in particular, prove Theorem 6 (the Prime Number Theorem). We begin, however, by proving the much simpler Theorem 7. Our proof of Theorems 6 and 7 depends upon the properties of a function there being a contribution log 2 from 2, 4, and 8, and a contribution log 3 from 3 and 9.If pmis the highest power of pnot exceeding x, log poccurs mtimes in Ψ(x).Also pmis the highest power of pwhich divides any number up to x, so that where U(x)is the least common multiple of all numbers up to x. We can also express Ψ(x)in the form † Throughout this chapter x(and yand t) are not necessarily integral. On the other hand, m, n, h, k, etc., are positive integers and p, as usual, is a prime. We suppose always that x): 1.