Theorems of Hoheisel and of Ingham

Abstract

The prime number theorem implies that p _n ~ nlogn, as n→∞, where p _n denotes the n^th prime. A related problem is to determine the size of the difference P_{n + 1} - P _n . The purpose of this chapter is to prove a theorem of Ingham’s which implies, in particular, that

$${p_{n + 1}} - {p_n} = 0\left( {p_n^{\frac{5} {8} + \varepsilon }} \right),\:as\quad n \to \infty$$

for every ε>0.