# On the size of the maximum of incomplete Kloosterman sums

Bonolis D. 2022. On the size of the maximum of incomplete Kloosterman sums. Mathematical Proceedings of the Cambridge Philosophical Society. 172(3), 563–590.

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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).

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Date Published

2022-05-01

Journal Title

Mathematical Proceedings of the Cambridge Philosophical Society

Publisher

Cambridge University Press

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

Issue

3

Page

563 - 590

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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*. 2022;172(3):563-590. doi:10.1017/S030500412100030XBonolis, D. (2022). 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/S030500412100030XBonolis, Dante. “On the Size of the Maximum of Incomplete Kloosterman Sums.”

*Mathematical Proceedings of the Cambridge Philosophical Society*. Cambridge University Press, 2022. 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*, vol. 172, no. 3. Cambridge University Press, pp. 563–590, 2022.Bonolis, Dante. “On the Size of the Maximum of Incomplete Kloosterman Sums.”

*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 172, no. 3, Cambridge University Press, 2022, pp. 563–90, doi:10.1017/S030500412100030X.**All files available under the following license(s):**

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arXiv 1811.10563