Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$

Let $p_1, \dotsc, p_k$ be primes not exceeding $x$ ($k \geqslant 2$), and define the additive Mertens sum \[ S_k(x) = \sum_{p_1 \leqslant x} \cdots \sum_{p_k \leqslant x} \frac{1}{p_1 + \dotsm + p_k}. \] In contrast to Tenenbaum's generalized (multiplicative) Mertens sum, whose leading term has or