The arcsine law on divisors in arithmetic progressions modulo prime powers
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THE ARCSINE LAW ON DIVISORS IN ARITHMETIC PROGRESSIONS MODULO PRIME POWERS B. FENG1,† and J. WU2,∗ 1
School of Mathematics and Statistics, Yangtze Normal University, Fuling, Chongqing 408100, China e-mail: [email protected] 2
CNRS LAMA 8050 Universit´ e Paris-Est Cr´ eteil, 94010 Cr´ eteil cedex, France e-mail: [email protected]
(Received March 12, 2020; revised August 13, 2020; accepted August 20, 2020)
Abstract. Let x → ∞ 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 exp{( 14 − ε)(log2 x)2 }, where ε is an arbitrarily small positive number. We show that in the case of a prime-power modulus (q := p with p a fixed odd prime and ∈ N) the arcsine law on divisors holds in arithmetic progressions for q ≤ x15/52−ε .
1. Introduction For each positive integer n, denote by τ (n) the number of divisors of n and define the random variable Dn to take the value (log d)/ log n, as d runs through the set of the divisors of n, with the uniform probability 1/τ (n). The distribution function Fn of Dn is given by (1.1)
Fn (t) := Prob(Dn t) =
1 τ (n)
1
(0 t 1).
d|n, dnt
Deshouillers, Dress and Tenenbaum ([4] or [10, Theorem II.6.7]) proved that the Ces` aro means of Fn converge uniformly to the arcsine law. More pre∗ Corresponding
author. paper was written when the first author visited LAMA 8050 de l’Universit´ e ParisEst Cr´ eteil 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). Key words and phrases: Selberg–Delange method, arcsine law, arithmetic progression. Mathematics Subject Classification: 11N37. † This
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2
B. FENG and J. WU
cisely, the asymptotic formula 1 √ 1 2 Fn (t) = arcsin t + O √ x π log x
(1.2)
nx
holds uniformly for x 2 and 0 t 1 and the error term in (1.2) is optimal. Various cases of (1.2) have been investigated by different authors. In particular, Cui and Wu [3] and Cui, L¨ u and Wu [2] considered generalizations of (1.2) to the short interval case; and Feng and Wu [6] showed that the average distribution of divisors over integers representable as the sum of two squares converges to the beta law. Based on Cui and Wu’s method [3], Feng [5] studied the analogue of (1.2) for arithmetic progressions. His result can be stated as follows: Let a and q be integers such that (a, q) = 1, and suppose that q is not an “exceptional modulus”. Then for any ε ∈ (0, 41 ) we have √log q √ 1 e 2 (1.3) Fn (t) = arcsin t + Oε √ (x/q) π log x nx n≡a(mod q)
uniformly for 0 t 1, x 2 and q exp{( 41 − ε)(log2 x)2 }, where log2 := log log. The aim of this paper is to improve the result above in the case of prime power modulus. Our result is as follows. Theorem 1.1. Let q := p with
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