A Family of New Formulas for the Euler-Mascheroni Constant

TL;DR

This paper introduces a new family of formulas for the Euler-Mascheroni constant using the E-Harmonic function, aiming to aid in understanding its properties and potential irrationality.

Contribution

It presents novel formulas and approximations for the Euler-Mascheroni constant derived from the newly defined E-Harmonic function.

Findings

Two new formulas for the Euler-Mascheroni constant

A family of formulas derived from the E-Harmonic function

Potential tools for investigating the constant's irrationality

Abstract

We introduce and prove several new formulas for the Euler-Mascheroni Constant. This is done through the introduction of the defined E-Harmonic function, whose properties, in this paper, lead to two novel formulas, alongside a family of formulas. While the paper does introduce many new approximations, it does not exhaust the possibilities of the E-Harmonic function but provides a strong first dive into its natural conclusions. We hope that the diversity of new formulas may provide stepping stones to a proof (or disproof) of the irrationality of the Euler-Mascheroni constant.