Properties of large prime divisors of numbers of the form \(p-1\) (Q2460496)

Let \(\mathcal Q_{x, \varepsilon}\) denote the set of primes \(p \leq x\) such that for all primes \(q\), with \(q \mid (p - 1)\), with \(\log q > \log x/\log\log x\), one has \[ (1 - \varepsilon)\log\log q \leq \omega(q - 1) \leq (1 + \varepsilon)\log\log q; \] here, \(\omega(n)\) denotes the number of distinct prime divisors of \(n\). The main result of the paper establishes that for any fixed \(\varepsilon > 0\) and \(x \to \infty\), almost all primes belong to \(\mathcal Q_{x, \varepsilon}\): \[ | \mathcal Q_{x, \varepsilon}| = (1 + o(1))x/\log x \quad \text{as } x \to \infty. \]