On the frequency of the digits of the largest prime factor of an integer

For an integer \(g \geqslant 2\) $g⩾2$ , we let \(S_g(n)\) $S_g(n)$ be the g-ary sum-of-digits function of a non negative integer n. This paper focus on statistical behavior of the sum-of-digits function restricted to the largest prime factor P(n) of the positive integer n. In particular, we provide asymptotic expansions for the numbers \(\# \{ n\leqslant x,\ \ S_g\left( P(n)\right) =d\}\) $\#\{n⩽x,S_g\left(P,(,n,)\right)=d\}$ for d close to \(\left( (g-1)/2\right) \log _{g}x\) $\left((,g,-,1,),/,2\right){log}_{g}x$ .