I am attemping to find asymptotics of

$$\sum_{p \leq n}\ln p \left( \sum_{k=1}^\infty \left(\left\{\frac{n}{p^k} \right\} - \left\{\frac{n-1}{(p-1)p^k} \right\} \right) - \left\{\frac{n-1}{p-1} \right\} \right) $$
where $\{ x\}$ denotes the *fractional part* of $x$ and $p$ denotes prime. On January 2016, Fedor Petrov consequently has shown

$$\sum_{p \leq n}\ln p\sum_{k=0}^\infty \left\{\frac{n-1}{(p-1)p^k} \right\} = (1-\gamma)n+O(n\epsilon_n),$$ where $\gamma$ denotes the Euler-Mascheroni constant and $\epsilon_n \to 0$. This leaves us with determining asymptotics for $$\sum_{p \leq n}\ln p\sum_{k=1}^\infty \left\{\frac{n}{p^k} \right\},$$ which I suspect is determined somewhere in literature since this is the fractional part discussed in Legendre's Theorem.