article published yes Dante Bonolis author 6A459894-5FDD-11E9-AF35-BB24E6697425 TiBr department 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). https://research-explorer.ista.ac.at/download/9364/10395/2021_MathProcCamPhilSoc_Bonolis.pdf application/pdfno Cambridge University Press2022 eng 0305-0041 1469-8064 1811.10563 00078442150000110.1017/S030500412100030X 1723563 - 590 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> D. Bonolis, Mathematical Proceedings of the Cambridge Philosophical Society 172 (2022) 563–590. Bonolis D. 2022. On the size of the maximum of incomplete Kloosterman sums. Mathematical Proceedings of the Cambridge Philosophical Society. 172(3), 563–590. 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>. 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. 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> 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>. 93642021-05-02T22:01:29Z2023-08-02T06:47:48Z