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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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.
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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:
  • 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:
  • 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:
  • MathSciNet review: 4419047