A note on Artin's constant

TL;DR

This paper introduces a novel approach to representing Artin's constant and related constants as ratios of infinite sums, offering new theoretical insights and computational possibilities.

Contribution

It proposes a new method to express Artin's constant and similar constants as ratios of infinite sums, extending to higher rank Artin's constants.

Findings

The approach provides a new theoretical representation of Artin's constant.

It applies successfully to Stephens' constant and higher rank Artin's constants.

The method has interesting computational and experimental implications.

Abstract

We suggest a new approach to Artin's constant that leads to its representation as an infinite sum divided by another infinite sum. The same approach works well for Stephens' constant and higher rank Artin's constants. The main results are theoretical but there are interesting experimental and computational aspects.