We establish asymptotic formulas for sums of reciprocals of primes in arithmetic progressions, generalizing recent results on multiple Mertens evaluations by Tenenbaum, Qi, and Hu. Specifically, for any fixed constant $K>0$, we derive asymptotic expansions for the sums $ \sum_{ \substack{ p_1\cdots p_n\leq x \\ p_i\equiv h_i \pmod{ m_i} \\ i=1,\dots, n}}\frac{ 1}{ p_1\cdots p_n} $ and the corresponding log-weighted sums. A key feature of our results is that the error terms hold \emph{ uniformly} for moduli satisfying $m_i \le (\log x)^K$, a range accessible via the Siegel-Walfisz theorem. Furthermore, we identify the coefficients of the asymptotic expansion with the Taylor series of the reciprocal Gamma function, $1/\Gamma(z)$, providing a structural explanation for the lower-order terms.
Categories: math.NT