Selberg筛权重与Kloosterman和的符号变化. II. 至多四个素因子的无平方因子模
Selberg sieve weights and sign changes of Kloosterman sums. II. Square-free moduli with at most four prime factors
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中文总结 AI 辅助
本文证明对每个符号，存在大量（\\(\gg X/\log X\\)）至多四个素因子的无平方因子模\\(q\\)，其归一化Kloosterman和取该符号，方法结合Selberg筛与等分布及均方估计。
中文摘要 AI 辅助
设\\(\Kl(1;q)\\)表示模\\(q\\)的归一化Kloosterman和。我们证明，对于每个符号，存在\\(\gg X/\log X\\)个无平方因子模\\(q\in(X,2X]\\)，其至多有四个素因子，使得\\(\Kl(1;q)\\)具有该符号。证明结合了Selberg筛与Kloosterman和的等分布及均方估计；第一部分中建立的移位筛权重的均匀估计控制了具有小素因子的模的贡献。
英文摘要
Let \(\Kl(1;q)\) denote the normalized Kloosterman sum modulo \(q\). We prove that, for each sign, there are \(\gg X/\log X\) square-free moduli \(q\in(X,2X]\) 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.
发表机构
- School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
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