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.
Best Answer
See "On the number of primes $p$ for which $p+a$ has a large prime factor." (Goldfeld, Mathematika 16 1969 23--27.) Using Bombieri-Vinogradov he proves, for a fixed integer $a$, that
$$\sum_{p \leq x} \sum_{\substack{ x^{1/2}< q \leq x \\ q | p+a}} \ln q = \frac{x}{2} + O\left(\frac{x \ln \ln x}{ \ln x} \right)$$
where the summation is over $p$ and $q$ prime. Note that this implies that the number of primes $p$ less than $x$ such that $p-1$ has a prime factor greater than $p^{1/2}$ is asymptotically at least $\frac{x}{2\ln x}$.