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Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
Abstract
Let Kl(1;q) denote the normalized Kloosterman sum modulo q. We prove that, for each sign, there are ≫ X/log X square-free moduli q∈(X,2X] with at most four prime factors for which Kl(1;q) has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for Kloosterman sums; the uniform estimate for shifted sieve weights established in Part~I controls the contribution of moduli having small prime factors.
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BibTeX
@misc{xiao2026selberg,
title = {{Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors}},
author = {Yixiu Xiao and Hongze Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24255},
eprint = {2609.24255},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24255},
}
RIS
TY - UNPB
TI - Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
AU - Yixiu Xiao
AU - Hongze Li
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24255
UR - https://arxiv.org/abs/2609.24255
AB - Let Kl(1;q) denote the normalized Kloosterman sum modulo q. We prove that, for each sign, there are ≫ X/log X square-free moduli q∈(X,2X] with at most four prime factors for which Kl(1;q) has that sign. The proof combines the Selberg sieve with equidistribution and mean-square estimates for Kloosterman sums; the uniform estimate for shifted sieve weights established in Part~I controls the contribution of moduli having small prime factors.
ER -