Distribution of quadratic residues of a fixed number without using Dedekind zeta function

This answer summarizes the above discussion: Extend $p mapsto left( frac{n}{p} right)$ to a multiplicative function $chi$ on the positive integers. By quadratic reciprocity, $chi$ is periodic modulo $4n$, and it is multiplicative by construction, so it is a character. We know that $L(1, chi) neq 0$. Thus, $lim_{s to 1^{+}} |log L(s,chi)| < infty$.

We compute:
$$log L(s, chi) = - sum log left( 1-frac{chi(p)}{p^s} right) = sum frac{chi(p)}{p^s} + O(1).$$

So $sum left( frac{n}{p} right)/p^s$ is bounded as $s to 1^{+}$. A little more work shows that the limit as $s to 1^{+}$ exists.

If you want to prove results like that $sum left( frac{n}{p} right)/p$ converges, or that $|sum_{p < N} left( frac{n}{p} right)| = o(N)$, then you need Tauberian methods, as discussed in any book on analytic number theory.