1 results
A Pólya–Vinogradov inequality for short character sums
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 64 / Issue 4 / December 2021
- Published online by Cambridge University Press:
- 02 December 2020, pp. 906-910
- Print publication:
- December 2021
-
- Article
- Export citation
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In this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume
$\chi $
to be a primitive character modulo q, 
$ \epsilon>0$
and 
$N\le q^{1-\gamma }$
, with 
$0\le \gamma \le 1/3$
. We prove that $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$with
$c=2/\pi ^2$
if 
$\chi $
is even and 
$c=1/\pi $
if 
$\chi $
is odd. The result is based on the work of Hildebrand and Kerr.
