  Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies
$\phi(p-1) \geq \phi(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that
the reverse inequality holds for a small positive proportion of the twin
primes. That is, $p$ tends to have more primitive roots than does $p+2$. We
prove that Dickson's conjecture, which is much weaker than Bateman-Horn,
implies that the quotients $\frac{\phi(p+1)}{\phi(p-1)}$, as $p,p+2$ range over
the twin primes, are dense in the positive reals. We also establish several
Schinzel-type theorems, some of them unconditional, about the behavior of
$\frac{\phi(p+1)}{\phi(p)}$ and $\frac{\sigma(p+1)}{\sigma(p)}$, in which
$\sigma$ denotes the sum-of-divisors function.
