Plus and minus logarithms and Amice transform

Abstract

We give a new description of Pollack's plus and minus $p$-adic logarithms $\log_p^\pm$ in terms of distributions. In particular, if $\mu_\pm$ denote the pre-images of $\log_p^\pm$ under the Amice transform, we give explicit formulae for the values $\mu_\pm(a+p^n\mathbb{Z}_p)$ for all $a\in \mathbb{Z}_p$ and all integers $n\ge1$. Our formulae imply that the distribution $\mu_-$ agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.