Multiple Mertens theorems for arithmetic progressions

Abstract

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, $1/Γ(z)$, providing a structural explanation for the lower-order terms.