= p-adic zeta pseudomeasure
{title2=$\int x^k\,d\zeta_p=(1-p^{k-1})\zeta(1-k)\quad(k\ge2\text{ even})$}
On $G=\mathbb Z_p^\times/\{\pm1\}$, this element of the total quotient ring of the <Iwasawa algebra> becomes an integral measure after multiplication by every difference $[a]-[1]$. Its even moments interpolate the displayed Euler-corrected zeta values. Its trivial-character component has a pole, so the pseudomeasure itself must not be treated as an element of the integral group algebra. Multiplication by the <augmentation ideal> removes the pole.
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