p-adic zeta pseudomeasure ($\int x^{k}d{\zeta }_p=(1-p^{k-1})\zeta (1-k)(k\ge 2\text{ even})$)

On $G={\mathbb{Z}}_p^{\times }/\{\pm 1\}$, 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.