On the normal number of prime factors of φ(n) subject to certain congruence conditions

JOURNAL OF NUMBER THEORY

卷:160

Article

Mkaouar, M.¹  Wannes, W.¹ 

[1] Fac Sci Sfax, Sfax 3000, Tunisia

关键词: Sum-of-digits function;    Erdos-Kac theorem;    Normal order;    Euler function;   

DOI  :  10.1016/j.jnt.2015.09.009

来源: Elsevier

【 摘 要 】

Let q >= 2 be an integer and S-q(n) denote the sum of the digits in base q of the positive integer n. It is proved that for every real number a and beta with alpha < beta, lim(x ->+infinity) 1/x #{n <= x : alpha <= v((phi(n)) - 1/2b (log log n)(2)/ 1/root 3h (log log n)(3/2) <= beta = 1/root 2 pi integral(beta)(alpha) e(-t2/2) dt, where v(n) is either (omega) over tilde (n) or (Omega) over tilde (n), the number of distinct prime factors and the total number of prime factors p of a positive integer n such that S-q(p) a mod b (a, b is an element of Z, b >= 2). This extends the results known through the work of P. Erdos and C. Pomerance, M.R. Murty and V.K. Murty to primes under digital constraint. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free