The Prime Number Theorem

Abstract

We shall now deduce, from the results of the last section and those of §13, that

$$\psi (x) = x + 0\left\{ {x\exp {{\left[ {\log x} \right]}^{\frac{1}{2}}}} \right\}$$

(1)

and from this the analogous result for π(x), which includes the prime number theorem. This is by no means the easiest way of proving the prime number theorem, but it is an instructive way. It is also very close to the method used by de la Vallée Poussin, though he worked with the function

$${\psi _1}(x) = \sum\limits_{n \le x} {(x - n)\Lambda (n)}$$

instead of the function φ(x).