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 $\\phi(p-1) \\geq \\phi(p+1)$ 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.