= Amice transform
{c}
{title2=$\mathcal A_\mu(T)=\int_{\mathbb Z_p}(1+T)^x\,d\mu(x)$}
The coefficient of $T^n$ is $\int\binom{x}{n}\,d\mu(x)$. The <Mahler expansion of continuous p-adic functions> identifies $I$-valued <p-adic measures> on $\mathbb Z_p$ with $I[[T]]$. If $D=(1+T)d/dT$, then $D^k\mathcal A_\mu(0)=\int x^k\,d\mu(x)$. A measure supported on $\mathbb Z_p^\times$ has $\sum_{\zeta^p=1}\mathcal A_\mu(\zeta(1+T)-1)=0$, because the sum of $\zeta^x$ is zero for $p\nmid x$.
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