论文标题
在算术进行中的无方整数以平滑模量
Squarefree Integers in Arithmetic Progressions to Smooth Moduli
论文作者
论文摘要
令$ε> 0 $足够小，让$ 0 <η<1/522 $。我们表明，如果$ x $在$ε$方面足够大，那么对于任何无平方的整数$ q \ q \ leq x^{196/261-ε} $，即$ x^η$ -smooth，一个人可以获得一个渐近公式，并获得带有电力误差术语的无功能误差术语的差异，$ pmod $ a $ pmod $ pmod pmod pmod pmod pmod pmod =在无方面的情况下，平滑模量在先前的Nunes工作上有所改善，其中$ 196/261 = 0.75096 ... $被$ 25/36 = 0.69 \ bar {4} $取代。这也为正密度模量建立了一个分布水平，该模量均以Hooley的结果改善。我们更普遍地表明，可以打破$ x^{3/4} $ - 密度为1集的$ x^η$ -Smooth Moduli $ q $的屏障（没有SquareFree条件）。我们的证明吸引了由于heath-brown而导致的van der corput方法的$ q $ - 动物，以将任务减少到估计某些Kloosterman型完全指数级的相关性，以估算Modulo Prime prime prime prime的相关性。在主要案例中，我们通过对这些完整总和的共同体处理获得省电的限制，而在较高的Prime Power Case中，我们使用$ p $ -ADIC方法建立了这种节省。
Let $ε> 0$ be sufficiently small and let $0 < η< 1/522$. We show that if $X$ is large enough in terms of $ε$ then for any squarefree integer $q \leq X^{196/261-ε}$ that is $X^η$-smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod{q}$, with $(a,q) = 1$. In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096...$ was replaced by $25/36 = 0.69\bar{4}$. This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$-barrier for a density 1 set of $X^η$-smooth moduli $q$ (without the squarefree condition). Our proof appeals to the $q$-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using $p$-adic methods.
