2019/09/24 by Tianfang Qi, Qi, Tianfang, Su Hu +1 · 1 citation
Mathematics · #11N05 #11N37 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Mathematical functions and polynomials #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1909.10930
openalex publication_date 2019/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Mertens' first theorem gives us the following asymptotic formula ∑_\substackp≤ x
p~prime(lnp)/(p)=lnx+O(1), and the Mertens' second theorem indicates that there exists a constant B≈ 0.261, named the Mertens constant, such that ∑_\substackp≤ x
p~prime(1)/(p)=ln(lnx)+B+O((1)/(lnx)). In this paper, by using the Abel summation formula and Dirichlet's hyperbola method, we extend them to multiple cases.