Multiple Mertens theorems for arithmetic progressions

TL;DR

This paper derives uniform asymptotic formulas for sums of reciprocals of primes in arithmetic progressions, extending previous results and linking the coefficients to the Taylor series of 1/Γ(z).

Contribution

It generalizes multiple Mertens theorems to arithmetic progressions with uniform error terms for moduli up to log powers, connecting asymptotics to the reciprocal Gamma function.

Findings

Asymptotic formulas valid for moduli up to (log x)^K

Coefficients match the Taylor series of 1/Γ(z)

Uniform error bounds via Siegel-Walfisz theorem

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