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

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Let q≥2

q \geq 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

α < β

\begin{matrix} \lim_{x ⟶ + \infty} \frac{1}{x} ♯ {n \leq x : α \leq \frac{v (φ (n)) - \frac{1}{2 b} {(\log \log n)}^{2}}{\frac{1}{\sqrt{3} b} {(\log \log n)}^{\frac{3}{2}}} \leq β} \\ = \frac{1}{\sqrt{2 π}} \int_{α}^{β} e^{- \frac{t^{2}}{2}} d t, \end{matrix}

where v(n)

v (n)

is either

\tilde{ω} (n)

\tilde{Ω} (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 \mod b

(a,b∈Z

a, b \in Z

, b≥2

b \geq 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.