$\zeta(-1)$ and $p$-adic numbers

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I have only a lay person's understanding of p-adic numbers, where a fraction may have an 'infinitely large' p-adic number representation. That got me thinking about $\zeta(-1)$= -1/12 being equal to the sum of the natural numbers. Is there a way to understand the sum of the natural numbers converging to -1/12 in the p-adic sense?