---
_id: '9364'
abstract:
- lang: eng
  text: 'Let t : Fp → C be a complex valued function on Fp. A classical problem in
    analytic number theory is bounding the maximum M(t) := max 0≤H<p ∣ 1/√p ∑ 0≤n<H
    t (n) ∣ of the absolute value of the incomplete sums(1/√p)∑0≤n<H t (n). In this
    very general context one of the most important results is the Pólya–Vinogradov
    bound M(t)≤IIˆtII∞ log 3p, where ˆt : Fp → C is the normalized Fourier transform
    of t. In this paper we provide a lower bound for certain incomplete Kloosterman
    sums, namely we prove that for any ε > 0 there exists a large subset of a ∈ F×p
    such that for kl a,1,p : x → e((ax+x) / p) we have M(kla,1,p) ≥ (1−ε/√2π + o(1))
    log log p, as p→∞. Finally, we prove a result on the growth of the moments of
    {M (kla,1,p)}a∈F×p. 2020 Mathematics Subject Classification: 11L03, 11T23 (Primary);
    14F20, 60F10 (Secondary).'
acknowledgement: I am most thankful to my advisor, Emmanuel Kowalski, for suggesting
  this problem and for his guidance during these years. I also would like to thank
  Youness Lamzouri for informing me about his work on sum of incomplete Birch sums
  and Tal Horesh for her suggestions on a previous version of the paper. Finally,
  I am very grateful to the anonymous referee for their careful reading of the manuscript
  and their valuable comments.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Dante
  full_name: Bonolis, Dante
  id: 6A459894-5FDD-11E9-AF35-BB24E6697425
  last_name: Bonolis
citation:
  ama: Bonolis D. On the size of the maximum of incomplete Kloosterman sums. <i>Mathematical
    Proceedings of the Cambridge Philosophical Society</i>. 2022;172(3):563-590. doi:<a
    href="https://doi.org/10.1017/S030500412100030X">10.1017/S030500412100030X</a>
  apa: Bonolis, D. (2022). On the size of the maximum of incomplete Kloosterman sums.
    <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. Cambridge
    University Press. <a href="https://doi.org/10.1017/S030500412100030X">https://doi.org/10.1017/S030500412100030X</a>
  chicago: Bonolis, Dante. “On the Size of the Maximum of Incomplete Kloosterman Sums.”
    <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. Cambridge
    University Press, 2022. <a href="https://doi.org/10.1017/S030500412100030X">https://doi.org/10.1017/S030500412100030X</a>.
  ieee: D. Bonolis, “On the size of the maximum of incomplete Kloosterman sums,” <i>Mathematical
    Proceedings of the Cambridge Philosophical Society</i>, vol. 172, no. 3. Cambridge
    University Press, pp. 563–590, 2022.
  ista: Bonolis D. 2022. On the size of the maximum of incomplete Kloosterman sums.
    Mathematical Proceedings of the Cambridge Philosophical Society. 172(3), 563–590.
  mla: Bonolis, Dante. “On the Size of the Maximum of Incomplete Kloosterman Sums.”
    <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 172,
    no. 3, Cambridge University Press, 2022, pp. 563–90, doi:<a href="https://doi.org/10.1017/S030500412100030X">10.1017/S030500412100030X</a>.
  short: D. Bonolis, Mathematical Proceedings of the Cambridge Philosophical Society
    172 (2022) 563–590.
date_created: 2021-05-02T22:01:29Z
date_published: 2022-05-01T00:00:00Z
date_updated: 2023-08-02T06:47:48Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1017/S030500412100030X
external_id:
  arxiv:
  - '1811.10563'
  isi:
  - '000784421500001'
file:
- access_level: open_access
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  creator: cchlebak
  date_created: 2021-12-01T14:01:54Z
  date_updated: 2021-12-01T14:01:54Z
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file_date_updated: 2021-12-01T14:01:54Z
has_accepted_license: '1'
intvolume: '       172'
isi: 1
issue: '3'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 563 - 590
publication: Mathematical Proceedings of the Cambridge Philosophical Society
publication_identifier:
  eissn:
  - 1469-8064
  issn:
  - 0305-0041
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the size of the maximum of incomplete Kloosterman sums
tmp:
  image: /images/cc_by.png
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  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
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volume: 172
year: '2022'
...