Let $$x\to\infty$$ be a parameter. Feng [5] proved that the Deshouillers–Dress–Tenenbaum’s arcsine law on divisors of the integers less than x also holds in arithmetic progressions for ``non-exceptional moduli" $$q\leqslant\exp\{(\frac{1}{4}-\varepsilon)(\log_2 x)^2\}$$ , where $$\varepsilon$$ is an arbitrarily small positive number. We show that in the case of a prime-power modulus ( $$q:=\mathfrak{p}^{\varpi}$$ with $$\mathfrak{p}$$ a fixed odd prime and $$\varpi\in \mathbb{N}$$ ) the arcsine law on divisors holds in arithmetic progressions for $$q\le x^{15/52-\varepsilon}$$ .

中文翻译：

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等差数列模素数幂中除数的反正弦定律

让 $$x\to\infty$$ 成为一个参数。Feng [5] 证明了 Deshouillers-Dress-Tenenbaum 的关于小于 x 的整数的除数的反正弦定律也适用于“非异常模”的等差级数 $$q\leqslant\exp\{(\frac{1} {4}-\varepsilon)(\log_2 x)^2\}$$ ，其中 $$\varepsilon$$ 是一个任意小的正数。我们证明在素数幂模数 ( $$q: =\mathfrak{p}^{\varpi}$$ 与 $$\mathfrak{p}$$ 一个固定的奇素数和 $$\varpi\in \mathbb{N}$$ ）除数的反正弦定律在算术中成立$$q\le x^{15/52-\varepsilon}$$ 的级数。