Multiple Mertens evaluations

TL;DR

This paper extends Mertens' theorems to multiple cases using Abel summation and Dirichlet's hyperbola method, providing generalized asymptotic formulas for sums over primes.

Contribution

It introduces a generalization of Mertens' theorems to multiple variables using advanced summation techniques.

Findings

Extended Mertens' theorems to multiple cases.

Derived asymptotic formulas for prime sums in multiple variables.

Utilized Abel summation and Dirichlet's hyperbola method.

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.