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

Abstract

Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $\varphi (p-1)\ge \varphi (p+1)$ 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 $\frac{\varphi (p+1)}{\varphi (p-1)}$, 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 $\frac{\varphi (p+1)}{\varphi (p)}$ and $\frac{\sigma (p+1)}{\sigma (p)}$, in which $\sigma$ denotes the sum-of-divisors function.

16 pages