Consider the equation
$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$
``proved'' by Ramanujan. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\infty} n^{-s}$ for $\Re(s) > 1$, and $\zeta(s)$ is defined by analytic continuation elsewhere.

I seem to remember being told once that this equation <i>was</i> true in the $p$-adic integers. However, on a moment's reflection this is clearly false; the infinite series does not converge in any $\mathbb{Q}_p$.

Is there any argument that an amended version of Ramanujan's statement is true $p$-adically, which does not imitate the usual arguments for $\mathbb{R}$? Is it ``obvious'' that the denominator should only be divisible by the primes 2 and 3?

Thank you!