Burgess inequality

In analytic number theory, the Burgess inequality is an inequality that provides an upper bound for character sums
where is a Dirichlet character modulo a cube free that is not the principal character.
The inequality was proven in 1963 along with a series of related inequalities, by the British mathematician David Allan Burgess. It provides a better estimate for small character sums than the Pólya–Vinogradov inequality from 1918. More recent results have led to refinements and generalizations of the Burgess bound.

A number is called cube free if it is not divisible by any cubic number except. Define with and.
Let be a Dirichlet character modulo that is not a principal character. For two, define the character sum
If either is cube free or, then the Burgess inequality holds
for some constant.