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

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

where ${\displaystyle \chi }$ is a Dirichlet character modulo a cube free ${\displaystyle p\in \mathbb {N} }$ that is not the principal character ${\displaystyle \chi _{0}}$.

The inequality was proven in 1963 along with a series of related inequalities, by the British mathematician David Allan Burgess.^[1] 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.^[2]

A number is called cube free if it is not divisible by any cubic number ${\displaystyle x^{3}}$ except ${\displaystyle \pm 1}$. Define ${\displaystyle r\in \mathbb {N} }$ with ${\displaystyle r\geq 2}$ and ${\displaystyle \varepsilon >0}$.

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

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

If either ${\displaystyle p}$ is cube free or ${\displaystyle r\leq 3}$, then the Burgess inequality holds^[3][4]

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

for some constant ${\displaystyle C_{r,\varepsilon }}$.

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