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

## TL;DR

This paper explores the distribution of totient and sum-of-divisors function quotients for twin primes, proving density results under Dickson's conjecture and establishing Schinzel-type theorems, some unconditionally.

## Contribution

It demonstrates that Dickson's conjecture implies the density of certain quotients for twin primes and proves new Schinzel-type theorems for these functions.

## Key findings

- Quotients of totient functions are dense in positive reals under Dickson's conjecture.
- Several Schinzel-type theorems about totient and sum-of-divisors ratios are established.
- Some results are proven unconditionally, strengthening the understanding of prime-related function behavior.

## 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.