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Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function
Journal of Number Theory ( IF 0.7 ) Pub Date : 2020-03-01 , DOI: 10.1016/j.jnt.2019.08.003
Stephan Ramon Garcia , Florian Luca , Kye Shi , Gabe Udell

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.

中文翻译:

孪生素数的原始根偏差 II:totient 商和除数和函数的 Schinzel 型定理

Garcia、Kahoro 和 Luca 表明,对于大多数孪生素数对 $p,p+2$,Bateman-Horn 猜想意味着 $\phi(p-1)\geq\phi(p+1)$反向不等式适用于小比例的孪生素数。也就是说,$p$ 往往比 $p+2$ 具有更多的原始根。我们证明比 Bateman-Horn 弱得多的 Dickson 猜想意味着商 $\frac{\phi(p+1)}{\phi(p-1)}$,如 $p,p+2$在孪生素数的范围内,在正实数中是密集的。我们还建立了几个 Schinzel 型定理,其中一些是无条件的,关于 $\frac{\phi(p+1)}{\phi(p)}$ 和 $\frac{\sigma(p+1) 的行为}{\sigma(p)}$,其中 $\sigma$ 表示除数和函数。
更新日期:2020-03-01
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