TITLE: Is Gauss sum a p-adic measure?
QUESTION [7 upvotes]: Let $\Gamma$ be Galois group of cyclotomic $\mathbb{Z}_p$ extension over $\mathbb{Q}$. Consider the function $G$ which sends each finite order character $\chi$ of $\Gamma$ to the Gauss sum $G(\chi)$, view $G(\chi)$ as an element of $\overline{\mathbb{Q}}_p$. My question is if $G$ is a $p$-adic measure on $\Gamma$?

REPLY [4 votes]: No. The Gauss sum is not a $p$-adic measure. One cheap way to see this is as follows: if $\chi$ has conductor $p^n$, the $p$-adic valuation of $G(\chi)$ is $n/2$. But if $\mu$ is a measure, the asymptotics of $\int \chi \mathrm{d}\mu$ for  $\chi$ of increasing $p$-power conductor, are governed by the $\lambda$ and $\mu$ invariants of $\mu$, and in particular the valuations of these numbers must tend to a limit -- they cannot tend to $\infty$.