---
_id: '10711'
abstract:
- lang: eng
  text: In this paper, we investigate the distribution of the maximum of partial sums
    of families of  m -periodic complex-valued functions satisfying certain conditions.
    We obtain precise uniform estimates for the distribution function of this maximum
    in a near-optimal range. Our results apply to partial sums of Kloosterman sums
    and other families of  ℓ -adic trace functions, and are as strong as those obtained
    by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular,
    we improve on the recent work of the third author for Birch sums. However, unlike
    character sums, we are able to construct families of  m -periodic complex-valued
    functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality
    is sharp.
acknowledgement: We would like to thank the anonymous referees for carefully reading
  the paper and for their remarks and suggestions.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Pascal
  full_name: Autissier, Pascal
  last_name: Autissier
- first_name: Dante
  full_name: Bonolis, Dante
  id: 6A459894-5FDD-11E9-AF35-BB24E6697425
  last_name: Bonolis
- first_name: Youness
  full_name: Lamzouri, Youness
  last_name: Lamzouri
citation:
  ama: Autissier P, Bonolis D, Lamzouri Y. The distribution of the maximum of partial
    sums of Kloosterman sums and other trace functions. <i>Compositio Mathematica</i>.
    2021;157(7):1610-1651. doi:<a href="https://doi.org/10.1112/s0010437x21007351">10.1112/s0010437x21007351</a>
  apa: Autissier, P., Bonolis, D., &#38; Lamzouri, Y. (2021). The distribution of
    the maximum of partial sums of Kloosterman sums and other trace functions. <i>Compositio
    Mathematica</i>. Cambridge University Press. <a href="https://doi.org/10.1112/s0010437x21007351">https://doi.org/10.1112/s0010437x21007351</a>
  chicago: Autissier, Pascal, Dante Bonolis, and Youness Lamzouri. “The Distribution
    of the Maximum of Partial Sums of Kloosterman Sums and Other Trace Functions.”
    <i>Compositio Mathematica</i>. Cambridge University Press, 2021. <a href="https://doi.org/10.1112/s0010437x21007351">https://doi.org/10.1112/s0010437x21007351</a>.
  ieee: P. Autissier, D. Bonolis, and Y. Lamzouri, “The distribution of the maximum
    of partial sums of Kloosterman sums and other trace functions,” <i>Compositio
    Mathematica</i>, vol. 157, no. 7. Cambridge University Press, pp. 1610–1651, 2021.
  ista: Autissier P, Bonolis D, Lamzouri Y. 2021. The distribution of the maximum
    of partial sums of Kloosterman sums and other trace functions. Compositio Mathematica.
    157(7), 1610–1651.
  mla: Autissier, Pascal, et al. “The Distribution of the Maximum of Partial Sums
    of Kloosterman Sums and Other Trace Functions.” <i>Compositio Mathematica</i>,
    vol. 157, no. 7, Cambridge University Press, 2021, pp. 1610–51, doi:<a href="https://doi.org/10.1112/s0010437x21007351">10.1112/s0010437x21007351</a>.
  short: P. Autissier, D. Bonolis, Y. Lamzouri, Compositio Mathematica 157 (2021)
    1610–1651.
corr_author: '1'
date_created: 2022-02-01T08:10:43Z
date_published: 2021-06-28T00:00:00Z
date_updated: 2024-10-21T06:02:06Z
day: '28'
department:
- _id: TiBr
doi: 10.1112/s0010437x21007351
external_id:
  arxiv:
  - '1909.03266'
  isi:
  - '000667289300001'
intvolume: '       157'
isi: 1
issue: '7'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1909.03266
month: '06'
oa: 1
oa_version: Preprint
page: 1610-1651
publication: Compositio Mathematica
publication_identifier:
  eissn:
  - 1570-5846
  issn:
  - 0010-437X
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: The distribution of the maximum of partial sums of Kloosterman sums and other
  trace functions
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 157
year: '2021'
...