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

Daoyi Peng, Hao Liu

Let $p_1,\dots ,p_k$ be primes not exceeding $x$ ($k⩾2$), and define the additive Mertens sum \[ S_k(x) = \sum_{p_1 \leqslant x} \cdots \sum_{p_k \leqslant x} \frac{1}{…