Linear and algebraic independence of Generalized Euler-Briggs constants

TL;DR

This paper investigates the algebraic independence and linear relations among generalized Euler-Briggs constants, establishing their infinite dimensionality over number fields and providing lower bounds on the dimension of the space they generate.

Contribution

It introduces new lower bounds on the dimension of spaces generated by generalized Euler-Briggs constants and proves their infinite dimensionality, advancing understanding of their algebraic independence.

Findings

Established non-trivial lower bounds on the dimension of spaces generated by these constants.

Proved the infinite dimensionality of the space spanned by generalized Euler-Briggs constants.

Studied linear and algebraic independence over algebraic numbers.

Abstract

Possible transcendental nature of Euler's constant $\gamma$ has been the focus of study for sometime now. One possible approach is to consider $\gamma$ not in isolation, but as an element of the infinite family of generalised Euler-Briggs constants. In a recent work \cite{GSS}, it is shown that the infinite list of generalized Euler-Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler-Briggs constants over number fields. More precisely, we obtain non-trivial lower bounds (see \thmref{pre} and \thmref{linear-ind}) on the dimension of these spaces and consequently establish the infinite dimensionality of the space spanned. Further, we study linear and algebraic independence of these constants over the field of all algebraic numbers.