It is well-known that cancellation in short character sums (eg Burgess' estimates) yields
bounds on the least quadratic nonresidue. Scant progress has been made on short …

We show that even mild improvements of the Pólya-Vinogradov inequality would imply
significant improvements of Burgess' bound on character sums. Our main ingredients are a …

A classical theorem of Paley asserts the existence of an infinite family of quadratic
characters whose character sums become exceptionally large. In this paper, we establish an …

We show in a quantitative way that any odd primitive character χ modulo q of fixed order g≥
2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to then …

For a primitive Dirichlet character $\chi $ modulo $ q $, we define $ M (\chi)=\max _ {t}|\sum _
{n\leq t}\chi (n)| $. In this paper, we study this quantity for characters of a fixed odd order …

Let χ (mod q) be a primitive Dirichlet character. In this paper, we prove a uniform upper
bound of the character sum∑ a∈ Aχ (a) over all proper generalized arithmetic progressions …

For a primitive Dirichlet character $\chi $ modulo $ q $, we define $ M (\chi)=\max_ {t}|\sum_
{n\leq t}\chi (n)| $. In this paper, we study this quantity for characters of a fixed odd order …

A heuristic in analytic number theory stipulates that sets of positive integers cannot
simultaneously be additively and multiplicatively structured. The practical verification of this …

For any given integer we prove the existence of infinitely many and characters of order such
that. We believe this bound to be the best possible. When the order is even, we obtain …

For a primitive Dirichlet character $\chi\pmod q $ we let\[M (\chi):=\frac {1}{\sqrt {q}}\max_
{1\leq t\leq q}\Big|\sum_ {n\leq t}\chi (n)\Big|.\] In this paper, we investigate the distribution of …