Multiple Mertens evaluations

Abstract

The Mertens' first theorem gives us the following asymptotic formula \begin{equation*} \sum_{\substack{p\leq x\\ p~prime}}\frac{lnp}{p}=lnx+O(1), \end{equation*} and the Mertens' second theorem indicates that there exists a constant $B\approx 0.261$, named the Mertens constant, such that \begin{equation*} \sum_{\substack{p\leq x\\ p~prime}}\frac{1}{p}=ln(lnx)+B+O\left(\frac{1}{lnx}\right). \end{equation*} In this paper, by using the Abel summation formula and Dirichlet's hyperbola method, we extend them to multiple cases.