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

Abstract

Let q≥2 be an integer and Sq(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,limx⟶+∞⁡1x♯{n≤x:α≤v(φ(n))−12b(log⁡log⁡n)213b(log⁡log⁡n)32≤β}=12π∫αβe−t22dt, where v(n) is either ω˜(n) or Ω˜(n), the number of distinct prime factors and the total number of prime factors p of a positive integer n such that Sq(p)≡amodb (a,b∈Z, b≥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.