What are the best known density results and conjectures for primes *p* where *p - 1* has a large prime factor *q*, where by "large" I mean something greater than $\sqrt{p}$.

The most extreme case is that of a safe prime (Wikipedia entry), which is a prime *p* such that $(p - 1)/2$ is also a prime (the smaller prime is called a Sophie Germain prime). I believe it is conjectured (and not yet proved) that infinitely many safe primes exist, and that the density is roughly $c/\log^2 n$ for some constant $c$ (as it should be from a probabilistic model).

For the more general setting, where we are interested in the density of primes *p* for which *p - 1* has a large prime factor, the only general approach I am aware of is the prime number theorem for arithmetic progressions, and some of its strengthenings such as the Bombieri-Vinogradov theorem (conditional to the GRH), the (still open) Elliott-Halberstam conjecture, Chowla's conjecture on the first Dirichlet prime, and some partial results related to this conjecture. All of these deal with the existence of primes $p \equiv a \pmod q$ for arbitrary *q* and arbitrary *a* that is coprime to *q*.

My question: can we expect qualitatively better results for the situation where *q* is prime and $a = 1$? Also, I am not interested in specifying *q* beforehand, so the existence of a *p* such that there exists *any* large prime *q* dividing $p - 1$ would be great. References to existing conjectures, conditional results, and unconditional results would be greatly appreciated.