Summary: Let \(q \geq 2\) 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 \(\alpha\) and \(\beta\) with \(\alpha < \beta\), \[ \lim_{x \to +\infty} \frac{1}{x} \# \left\{n\leq x: \alpha \leq \frac{v(\varphi(n)) - \frac{1}{2 b}(\log\log n)^2}{\frac{1}{\sqrt{3} b}(\log\log n)^{\frac{3}{2}}} \leq \beta \right\} = \frac{1}{\sqrt{2 \pi}} \int_\alpha^\beta e^{- \frac{t^2}{2}} \,dt, \] where \(v(n)\) is either \(\widetilde{\omega}(n)\) or \(\widetilde{\Omega}(n)\), the number of distinct prime factors and the total number of prime factors \(p\) of a positive integer \(n\) such that \(S_q(p) \equiv a \bmod b\) \((a, b \in \mathbb{Z}\), \(b\geq 2\)). This extends the results known through the work of P. Erdős and C. Pomerance, M. Ram Murty and V. Kumar Murty to primes under digital constraint.