## Large odd order character sums and improvements of the Pólya-Vinogradov inequality

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- by Youness Lamzouri and Alexander P. Mangerel PDF
- Trans. Amer. Math. Soc.
**375**(2022), 3759-3793 Request permission

## Abstract:

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 $g\geq 3$. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character $\chi$ we have \begin{equation*} M(\chi )\ll _{\varepsilon } \sqrt {q}(\log q)^{1-\delta _g}(\log \log q)^{-1/4+\varepsilon }, \end{equation*} where $\delta _g ≔1-\frac {g}{\pi }\sin (\pi /g)$. This improves upon the works of Granville and Soundararajan [J. Amer. Math. Soc. 20 (2007), pp. 357–384] and of Goldmakher [Algebra Number Theory 6 (2012), pp. 123–163]. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that \begin{equation*} M(\chi ) \ll \sqrt {q} \left (\log _2 q\right )^{1-\delta _g} \left (\log _3 q\right )^{-\frac {1}{4}}\left (\log _4 q\right )^{O(1)}, \end{equation*} where $\log _j$ is the $j$-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of $\log _4 q$). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest.## References

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## Additional Information

**Youness Lamzouri**- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J1P3, Canada
- Address at time of publication: Institut Élie Cartan de Lorraine, Université de Lorraine, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France.
- MR Author ID: 804642
- Email: youness.lamzouri@univ-lorraine.fr
**Alexander P. Mangerel**- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada
- Address at time of publication: Department of Mathematical Sciences, Durham University, Durham, DH1 3LE, United Kingdom
- MR Author ID: 1141860
- Email: smangerel@gmail.com
- Received by editor(s): August 20, 2018
- Received by editor(s) in revised form: September 18, 2019
- Published electronically: March 4, 2022
- Additional Notes: The first author was partially supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 3759-3793 - MSC (2020): Primary 11L40
- DOI: https://doi.org/10.1090/tran/8607
- MathSciNet review: 4419047