2025/03/12 by Samuels, Charles L.
#11R27 #15A04 #46E15 #FOS: Mathematics #Number Theory (math.NT) #Primary 11R04 #Secondary 11G50
paper · doi:10.48550/arxiv.2503.09752
A 2009 article of Allcock and Vaaler explored the \mathbb Q-vector space \mathcal G := \mathbb Q^×/\mathbb Q^×tors, showing how to represent it as part of a function space on the places of \mathbb Q. We establish a representation theorem for the \mathbb R-vector space of \mathbb Q-linear maps from \mathcal G to \mathbb R, enabling us to classify extensions to \mathcal G of completely additive arithmetic functions. We further outline a strategy to construct \mathbb Q-linear maps from \mathcal G to \mathbb Q, i.e., elements of the algebraic dual of \mathcal G. Our results make heavy use of Dirichlet's S-unit Theorem as well as a measure-like object called a consistent map, first introduced by the author in previous work.