On the size of the maximum of incomplete Kloosterman sums

Bonolis D. 2021. On the size of the maximum of incomplete Kloosterman sums. Mathematical Proceedings of the Cambridge Philosophical Society., 1–28.


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Abstract
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).
Publishing Year
Date Published
2021-04-15
Journal Title
Mathematical Proceedings of the Cambridge Philosophical Society
Page
1-28
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eISSN
IST-REx-ID

Cite this

Bonolis D. On the size of the maximum of incomplete Kloosterman sums. Mathematical Proceedings of the Cambridge Philosophical Society. 2021:1-28. doi:10.1017/S030500412100030X
Bonolis, D. (2021). On the size of the maximum of incomplete Kloosterman sums. Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press. https://doi.org/10.1017/S030500412100030X
Bonolis, Dante. “On the Size of the Maximum of Incomplete Kloosterman Sums.” Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, 2021. https://doi.org/10.1017/S030500412100030X.
D. Bonolis, “On the size of the maximum of incomplete Kloosterman sums,” Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press, pp. 1–28, 2021.
Bonolis D. 2021. On the size of the maximum of incomplete Kloosterman sums. Mathematical Proceedings of the Cambridge Philosophical Society., 1–28.
Bonolis, Dante. “On the Size of the Maximum of Incomplete Kloosterman Sums.” Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2021, pp. 1–28, doi:10.1017/S030500412100030X.
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