On the order of finitely generated subgroups of \(\mathbb{Q}^*\pmod p\) and divisors of \(p-1\) (Q1912284)

Let \(a_1,\dots, a_r\) be multiplicatively independent non-zero integers, and let \(G_p\) be the multiplicative group that they generate modulo \(p\). It is not hard to show that if \(f(p)\) tends to infinity with \(p\), then \(\# G_p\geq (p/ f(p))^{r/(r+ 1)}\) for almost all \(p\). The paper improves this to \[ \# G_p\geq p^{r/(r+ 1)} \exp \Biggl\{{\log^\tau p\over \exp(f(p) \sqrt{\log\log p})}\Biggr\}, \] where \(\tau= (1- \log 2)/2> 0\). A number of related results are given. The key ingredient in the proofs is a delicate upper bound for the number of primes \(p\) up to \(x\) for which \(p- 1\) has a divisor in a specified range.