Hooley假设$R^*$下平方模数的大筛法
The large sieve for square moduli under Hooley's hypothesis $R^*$
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中文总结 AI 辅助
在Hooley假设$R^*$下，将平方模数大筛法在临界点$N=Q^3$的界改进为$Q^{1/2-1/134+\varepsilon}$，通过精确计算高斯和将模平方根和转化为Salié和，获得对所有模数有效的平方根抵消。
中文摘要 AI 辅助
设$S(Q,M,N,(a_n)):=\sum_{q\le Q}\sum_{(a,q)=1}|\sum_{M<n\le M+N}a_ne(an/q^2)|^2$为Zhao的平方模数大筛法和。在临界点$N=Q^3$处，由Baier和Zhao（2008）给出的最佳已知无条件界为$S\ll Q^{1/2+\varepsilon}N\sum |a_n|^2$，而猜想界为$Q^{\varepsilon}N\sum|a_n|^2$，且指数$\tfrac12$至今未被降低。我们证明，在Hooley的短Salié和假设$R^*$下——即对周期任意子区间上的$\sum_{x_1<n\le x_2}\big(\tfrac nc\big)e_c(a\bar n+bn)$具有平方根抵消——当$N=Q^3$时有$S\ll Q^{1/2-1/134+\varepsilon}N\sum|a_n|^2$。关键估计是分数$a/q^2$（$q\le Q$）在接近$b/r$的点$\alpha$的$Q^{-3}$邻域内的个数$P(\alpha)$的界：我们证明对每个模数$Q^{1/2+\varepsilon}\le r\le Q^{3/2}$，有$P(b/r+z)\ll(Q^{2/3}r^{-1/3}+Q^{1/4})Q^\varepsilon$，改进了Baier（2026）仅对$r=p,p^2$得到的界$Q^{9/16}r^{-1/8}$，并推广到所有模数。证明基于一个单一观察：区间$J$上的模平方根和$\sum_{n\in J}e_r(a\sqrt{jn})$，在补全并对每个模数精确计算二次高斯和之后，等于$r^{-1/2}$乘以长度为$r/|J|$的Salié和。因此假设$R^*$直接对所有模数给出这些和的平方根抵消，无需Weyl差分；相对于平凡界的节省是Weyl差分路线所得节省的平方。高斯和的计算（包括偶数模数和与模数有公因子的系数）被完整证明。本文与Claude（Anthropic）合作准备；第1.9节说明了各方的贡献。
英文摘要
Let $S(Q,M,N,(a_n))=\sum_{q\le Q}\sum_{(a,q)=1}|\sum_{M<n\le M+N}a_ne(an/q^2)|^2$ be Zhao's large sieve sum with square moduli. At the critical point $N=Q^3$ the best known unconditional bound, due to Baier and Zhao (2008), is $S\ll Q^{1/2+ε}N\sum|a_n|^2$, against the conjectured $Q^εN\sum|a_n|^2$, and the exponent $1/2$ has not been lowered since. We prove that, under Hooley's Hypothesis $R^*$ for short Salié sums -- square-root cancellation for $\sum_{x_1<n\le x_2}(n/c)e_c(a\bar n+bn)$ over arbitrary subintervals of a period -- one has $S\ll Q^{1/2-1/28+ε}N\sum|a_n|^2$ at $N=Q^3$. The key estimate is a bound for the number $P(α)$ of fractions $a/q^2$, $q\le Q$, within $Q^{-3}$ of a point $α$ near $b/r$: we show $P(b/r+z)\ll(Q^{2/3}r^{-1/3}+Q^{1/4})Q^ε$ for every modulus $Q^{1/2+ε}\le r\le Q^{3/2}$, improving Baier's bound $Q^{9/16}r^{-1/8}$ for $r=p,p^2$. The proof rests on one observation: a sum of modular square roots $\sum_{n\in J}e_r(a\sqrt{jn})$ over an interval $J$ is, after completion and an exact evaluation of quadratic Gauss sums at every modulus, $r^{-1/2}$ times a Salié sum of length $r/|J|$, so Hypothesis $R^*$ yields square-root cancellation directly, at every modulus, without Weyl differencing. The relation of $R^*$ to the conjecture of Friedlander and Iwaniec is made precise. The size of the saving is decided by the small moduli $r\le Q^{2/3}$, where the argument is unconditional: for very small moduli we use Cauchy--Schwarz over the residue classes modulo $r$ followed by the second-derivative test, and for $Q^{3/7}\le r\le Q^{2/3}$ an improvement of the complete-sum input to Baier's earlier treatment, proved in the second part of the paper, which also corrects two steps of that treatment. The paper was prepared in collaboration with Claude (Anthropic); Section 1.10 records what each of us contributed.
发表机构
- Ramakrishna Mission Vivekananda Educational and Research Institute（罗摩克里希那传教士维韦卡南达教育与研究机构）
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