Burgess Bound

In analytic number theory, the Burgess inequality (also called the Burgess bound) is an inequality that provides an upper bound for character sums
:$S_(N,H):=\sum\limits_ \chi(n)$
where $\chi$ is a Dirichlet character modulo a cube free $p\in\mathbb$ that is not the principal character $\chi_0$.

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 $x^3$ except $\pm 1$. Define $r\in \mathbb$ with $r\geq 2$ and $\varepsilon>0$.

Let $\chi$ be a Dirichlet character modulo $p\in\mathbb$ that is not a principal character. For two $N,H\in\mathbb$, define the character sum
:$S_(N,H):=\sum\limits_ \chi(n).$

If either $p$ is cube free or $r\leq 3$, then the Burgess inequality holds
:$, S_(N,H), \leq C_ H^q^$
for some constant $C_{r,\varepsilon}$.

References

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

Notes

Analytic number theory
Theorems in analytic number theory