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

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• 作者：M. Mkaouar ; ^{mohamed.mkaouar@fss.rnu.tn" class="auth_mail" title="E-mail the corresponding author} ; W. Wannes ^{walid.wannes@fss.rnu.tn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11A63 ; 11L03 ; 11N05
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：629-645
• 全文大小：328 K

文摘

Let q≥2$q\ge 2$ be an integer and S_q(n)$S_q(n)$ denote the sum of the digits in base q of the positive integer n. It is proved that for every real number α and β   with α<β$\alpha <\beta$, where v(n)$v(n)$ is either $\tilde{\omega }(n)$ or $\tilde{\mathrm{\Omega }}(n)$, the number of distinct prime factors and the total number of prime factors p of a positive integer n   such that $S_q(p)\equiv a\mathrm{mod}b$ (a,b∈Z$a,b\in \mathbb{Z}$, b≥2$b\ge 2$). This extends the results known through the work of P. Erdős and C. Pomerance, M.R. Murty and V.K. Murty to primes under digital constraint.