TITLE: A divergent series related to the number of divisors of of p-1
QUESTION [6 upvotes]: Let $d(n)$ denote the number of divisors of $n$. Is it known that the series
$$\sum_{p \text{ prime}} \frac{1}{d(p-1)}$$
diverges?
This would follow immediately from the Sophie Germain Conjecture. Indeed, if there are infinitely many primes of the form $2p+1$ ($p$ a prime), then infinitely many terms of the series are equal to $1/4$, so the series doesn't even satisfy the most basic requirement for convergence! So, surely there must be a direct proof?

REPLY [4 votes]: Answering my own question, because I totally overlooked the following ridiculous idea:
Obviously $d(n)\leq n$ for every $n$. Thus $d(p-1)\leq p-1 < p$, so $1/d(p-1) > 1/p$ and the divergence follows from the divergence of $\sum 1/p$ (if one is willing to assume that).