On the difference in values of the Euler totient function near prime arguments

Stephan Ramon Garcia, Florian Luca

Published 2017-06-01Version 1

It is known that the inequality $\phi(p-1) > \phi(p+1)$ holds for an overwhelming majority of twin prime pairs $p,p+2$ (at least $65\%$ and at most $99.53\%$) if one assumes the Bateman-Horn conjecture. We show that this bias disappears if only $p$ is assumed to be prime. In fact, we prove unconditionally that for each $\ell \geq 1$, the difference $\phi(p-\ell) - \phi(p+\ell)$ is positive for $50\%$ of primes $p$ and negative for $50\%$.