Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor

TL;DR

This paper computes two slowly converging sums involving the reciprocal of an integer and its greatest prime factor to high precision, transforming them into sums with Mertens' product and estimating remainders using Chebyshev's -function.

Contribution

It introduces a novel method to evaluate these sums precisely by transforming them and estimating remainders with advanced number theory functions.

Findings

Sums are computed to 13 decimal digits accuracy.

Transformation reduces slow convergence of original sums.

Estimates of remainders are achieved using Chebyshev's -function.

Abstract

Two sums over the inverse of the product of an integer n and its greatest prime factor G(n), are computed to first 13 decimal digits. These sums converge, but converge very slowly. They are transformed into sums involving Mertens' prime product with the remainder term which are estimated by means of Chebyshev's {\theta}-function.