Abstract
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 \(q\le x^{15/52-\varepsilon}\).
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This paper was written when the first author visited LAMA 8050 de l’Université Paris-Est Créteil during the academic year 2019-2020. He would like to thank the institute for the pleasant working conditions. This work is partially supported by NSF of Chongqing (Nos. cstc2018jcyjAX0540, cstc2019jcyj-msxmX0009 and cstc2019jcyj-msxm1651).
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Feng, B., Wu, J. The arcsine law on divisors in arithmetic progressions modulo prime powers. Acta Math. Hungar. 163, 392–406 (2021). https://doi.org/10.1007/s10474-020-01105-7
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DOI: https://doi.org/10.1007/s10474-020-01105-7