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arXiv · 1906.05927

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

Stephan Ramon Garcia·Florian Luca·Kye Shi·Gabe Udell
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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.

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Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe Udell. 2019-06-13. Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function. https://arxiv.org/abs/1906.05927

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