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Horizontal moments of Kloosterman sums and a density version of sieves
Abstract
Denote by Kl(a,n) the normalized Kloosterman sum modulo n. In this paper, we study the moments of |Kl(1,n)| on average over nleqslant X, and show that ∑nleqslant X|Kl(1,n)|asymp X(log X)8/3π−1. The lower bound is proven by considering a subfamily of moduli which factorize in a nice way such that vertical Sato--Tate distributions of Kloosterman sums (as variants of Katz) apply. The upper bound is based on a new density version of upper bound sieves, combining with averages of divisor functions in arithmetic progressions and a similar vertical Sato--Tate distribution.
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BibTeX
@misc{xi2026horizontal,
title = {{Horizontal moments of Kloosterman sums and a density version of sieves}},
author = {Ping Xi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11083},
eprint = {2610.11083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11083},
}
RIS
TY - UNPB
TI - Horizontal moments of Kloosterman sums and a density version of sieves
AU - Ping Xi
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11083
UR - https://arxiv.org/abs/2610.11083
AB - Denote by Kl(a,n) the normalized Kloosterman sum modulo n. In this paper, we study the moments of |Kl(1,n)| on average over nleqslant X, and show that ∑_nleqslant X|Kl(1,n)|asymp X(log X)^8/3π−1. The lower bound is proven by considering a subfamily of moduli which factorize in a nice way such that vertical Sato--Tate distributions of Kloosterman sums (as variants of Katz) apply. The upper bound is based on a new density version of upper bound sieves, combining with averages of divisor functions in arithmetic progressions and a similar vertical Sato--Tate distribution.
ER -