On the size of the maximum of incomplete Kloosterman sums Bonolis, Dante ddc:510 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). Cambridge University Press 2022 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/9364 https://research-explorer.ista.ac.at/download/9364/10395 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> eng info:eu-repo/semantics/altIdentifier/doi/10.1017/S030500412100030X info:eu-repo/semantics/altIdentifier/issn/0305-0041 info:eu-repo/semantics/altIdentifier/e-issn/1469-8064 info:eu-repo/semantics/altIdentifier/wos/000784421500001 info:eu-repo/semantics/altIdentifier/arxiv/1811.10563 https://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess