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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 thatthe reverse inequality holds for a small positive proportion of the twinprimes. That is, $p$ tends to have more primitive roots than does $p+2$. Weprove 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 overthe twin primes, are dense in the positive reals. We also establish severalSchinzel-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.

Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function