Generalized Mertens sums

TL;DR

This paper extends Mertens' classical estimate to provide an explicit remainder formula for the product of any fixed number of primes, enhancing understanding of prime reciprocal sums.

Contribution

It introduces a generalized formula for Mertens sums applicable to products of multiple primes, with explicit remainder terms.

Findings

Derived an explicit remainder formula for prime product sums.

Generalized Mertens estimate to arbitrary fixed prime counts.

Enhanced precision in prime reciprocal sum approximations.

Abstract

Extending a classical estimate of Mertens for the sum of the reciprocals of the first primes, we provide an explicit remainder formula for products of an arbitrary, but fixed, number of primes.