Multiple Mertens theorems for arithmetic progressions

2025/12/08 by Chen, Zhen, Luo, Junrong
Mathematics · #11N05 #11N13 #11N37 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.07336

openalex publication_date 2025/12/08 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

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 ∑_\substackp₁⋯ p_n≤ x p_i≡ h_i \pmodm_i i=1,…, n(1)/(p₁⋯ p_n) and the corresponding log-weighted sums. A key feature of our results is that the error terms hold uniformly for moduli satisfying m_i ≤ (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.

Citations

• Multiple Mertens evaluations
• Generalized Mertens sums

Related

• Multiple Mertens evaluations
• Partial sums of the Möbius function in arithmetic progressions assuming GRH
• On the constants in Mertens' theorems for primes in arithmetic progressions
• Generalized Mertens sums
• Asymptotic equivalents of partial sums of the reciprocals of prime numbers via the von Mangoldt function