Convergence of the Riemann zeta function in $\mathbb Q_p$

Question

Does the Riemann zeta function without p-Euler factor i.e. $\prod\limits_{\text{prime }q \not= p}\frac{1}{1-q^{-1}}$ converges in $\mathbb Q_p$?

Answer 1

Note that $$ \frac1{1-q^{-1}}=\frac q{q-1}. $$ This fraction is not in $\Bbb Z_p$ for all $q\equiv1\bmod p$ and by Dirichlet's famous theorem there are infinitely many such $q$.

This shows that the Euler factor is $p$-adically large for infinitely many $q$ and $p$-adically small for no $q$ ($p$ never divides the numerator).

Thus $\prod_{q\neq p}\frac1{1-q^{-1}}$ defines no $p$-adic number.