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- Title
Short Character Sums and the Pólya–Vinogradov Inequality.
- Authors
Mangerel, Alexander P
- Abstract
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 $$\begin{equation*} \max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q) \end{equation*}$$ then for any ɛ > 0 one may exhibit cancellation in partial sums of χ on the interval [1, t ] whenever |$t \gt q^{\varepsilon}$| , i.e. $$\begin{equation*} \sum_{n \leq t} \chi(n) = o_{q \rightarrow \infty}(t)\ \text{for all } t \gt q^{\varepsilon}. \end{equation*}$$ We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya–Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.
- Subjects
CHARACTER; MATHEMATICAL equivalence
- Publication
Quarterly Journal of Mathematics, 2020, Vol 71, Issue 4, p1281
- ISSN
0033-5606
- Publication type
Article
- DOI
10.1093/qmath/haaa031