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

$S_{\chi }(N,H):=\sum _{N+1\le n\le N+H}\chi (n)$

where $\chi$ is a Dirichlet character modulo a cube free $p\in \mathbb{\mathbb{N}}$ 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 $r\in \mathbb{\mathbb{N}}$ with $r\ge 2$ and $\epsilon >0$.

Let $\chi$ be a Dirichlet character modulo $p\in \mathbb{\mathbb{N}}$ that is not a principal character. For two $N,H\in \mathbb{\mathbb{N}}$, define the character sum

$S_{\chi }(N,H):=\sum _{N+1\le n\le N+H}\chi (n).$

If either $p$ is cube free or $r\le 3$, then the Burgess inequality holds

$|S_{\chi }(N,H)|\le C_{r,\epsilon }{H}^{1-1/r}{q}^{(r+1)/(4r^2)+\epsilon }$

for some constant $C_{r,\epsilon }$.

References

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