AI 中文总结
研究具有至多五个素因子的模的克洛斯特曼符号变化，基于解析估计和优化筛法，用几何半权重改进方法，使相关系数降低，证明归一化克洛斯特曼和每个符号出现次数\(\gg X / \log X\)，改进了前人结果。
AI 中文摘要
对于在区间\((X,2X]\)内且至多有五个素因子的无平方因子模\(q\)，我们证明归一化克洛斯特曼和\(\operatorname{Kl}(1;q)\)的每个符号出现\(\gg X / \log X\)次。这改进了张和钟最近关于至多六个素因子模的无条件结果。基于他们的解析估计和优化的塞尔伯格筛法，我们用几何半权重取代他们的截断除数惩罚。新权重保留了\(P_5\)排除阈值，是两个标准双参数截断除数权重的正线性组合，张 - 钟转移论证无需改变即可应用。转移后，相关逐点系数从\(5/16\)降至\(5/32\)，从而产生正的最终筛余量。
英文摘要
On square-free moduli $q\in(X,2X]$ having at most five prime factors, we prove that each sign of the normalized Kloosterman sum $\operatorname{Kl}(1;q)$ occurs $\gg X/\log X$ times. This improves the recent unconditional result of Zhang and Zhong for moduli with at most six prime factors. Building on their analytic estimates and optimized Selberg sieve, we replace their truncated divisor penalty by a geometric half-weight. The new weight retains the $P_5$ exclusion threshold and is a positive linear combination of two standard two-parameter truncated divisor weights, so the Zhang--Zhong transference argument applies without alteration. After transference, the relevant pointwise coefficient is reduced from $5/16$ to $5/32$, which yields a positive final sieve margin.
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