Let $P$ be the set of primes. Consider the function $f(n) = \sum_{p|n} \frac{1}{p}$, where $p$ ranges over prime divisors of $n$. For a fixed positive integer $k$, define $S_k = \{n \in \mathbb{N} : n = p_1 p_2 \dots p_k \text{ where } p_i \text{ are distinct primes and } p_i \equiv 1 \pmod 4\}$. Which of the following is true regarding the density of the set of integers $n$ that are products of $k$ distinct primes congruent to $1 \pmod 4$?