AI 中文总结
该研究证明了无平方因子整数上指数和的新上界估计，改进了此前带$N^ε$因子的结果，采用带素数区间限制的平方筛法等技术，推导了Hardy-Littlewood剖分劣弧上的对应估计。
AI 中文摘要
设$μ$为莫比乌斯函数，$e(t)=e^{2πit}$。我们证明：若$N\ge2$，$α\in\mathbb{R}$，$(a,q)=1$，且$|α-a/q|\le q^{-2}$，则有\\[\bigg|\sum_{n\le N}μ^2(n)e(αn)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \\]其中隐含常数为绝对常数，并推导了在$Q\le N^{1/2}$的整个范围内，Hardy--Littlewood剖分的劣弧上的对应估计。Schlage-Puchta[SP]和Tolev[T]的估计对$q$和$Q$有相同的依赖关系，但带有一个$N^{\varepsilon}$因子。证明使用了Heath-Brown的平方筛法，其中筛素数被限制在区间$(P,2P]$内，$P$可小至$\log N$的倍数；用有限Fejér控制函数代替截断傅里叶级数；并在特征和完备化后，利用素数的限制条件而非除数函数来计数表示数。
英文摘要
Let $μ$ be the Möbius function and $e(t)=e^{2πit}$. We prove that if $N\ge2$, $α\in\mathbb{R}$, $(a,q)=1$, and $|α-a/q|\le q^{-2}$, then \[\bigg|\sum_{n\le N}μ^2(n)e(αn)\bigg|\ll\left(\frac Nq+q\right)(\log 2N)^5, \] with an absolute implied constant, and we deduce the corresponding estimate on the minor arcs of the Hardy--Littlewood dissection throughout the range $Q\le N^{1/2}$. The estimates of Schlage-Puchta [SP] and of Tolev [T] have the same dependence on $q$ and $Q$ but carry a factor $N^{\varepsilon}$. The proof uses Heath-Brown's square sieve with sieving primes confined to an interval $(P,2P]$, where $P$ may be as small as a multiple of $\log N$; a finite Fejér majorant in place of a truncated Fourier series; and, after completion of the character sums, a count of representations that exploits the restriction on the primes in place of the divisor function.
Comments16 pages
