Burgess inequality

In analytic number theory, the Burgess inequality (also called the Burgess bound) 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.

Burgess inequality

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 .

References

• Henryk Iwaniec and Emmanuel Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004.

Notes