{"status":"public","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.","volume":172,"ddc":["510"],"scopus_import":"1","month":"05","publication":"Mathematical Proceedings of the Cambridge Philosophical Society","citation":{"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.","short":"D. Bonolis, Mathematical Proceedings of the Cambridge Philosophical Society 172 (2022) 563–590.","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>.","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>","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>.","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."},"isi":1,"publication_status":"published","doi":"10.1017/S030500412100030X","quality_controlled":"1","year":"2022","_id":"9364","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","external_id":{"isi":["000784421500001"],"arxiv":["1811.10563"]},"author":[{"id":"6A459894-5FDD-11E9-AF35-BB24E6697425","last_name":"Bonolis","first_name":"Dante","full_name":"Bonolis, Dante"}],"date_updated":"2023-08-02T06:47:48Z","has_accepted_license":"1","oa":1,"abstract":[{"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).","lang":"eng"}],"oa_version":"Published Version","date_published":"2022-05-01T00:00:00Z","department":[{"_id":"TiBr"}],"file":[{"access_level":"open_access","success":1,"file_id":"10395","file_name":"2021_MathProcCamPhilSoc_Bonolis.pdf","relation":"main_file","content_type":"application/pdf","date_created":"2021-12-01T14:01:54Z","file_size":334064,"date_updated":"2021-12-01T14:01:54Z","creator":"cchlebak","checksum":"614d2e9b83a78100408e4ee7752a80a8"}],"article_type":"original","publisher":"Cambridge University Press","type":"journal_article","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"intvolume":"       172","page":"563 - 590","issue":"3","day":"01","article_processing_charge":"Yes (via OA deal)","title":"On the size of the maximum of incomplete Kloosterman sums","file_date_updated":"2021-12-01T14:01:54Z","publication_identifier":{"eissn":["1469-8064"],"issn":["0305-0041"]},"date_created":"2021-05-02T22:01:29Z","language":[{"iso":"eng"}]}