Euler's Function on Products of Primes in Progressions

Abstract

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\phi (n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\le q\le 10$ and for $q=12,14$, the generalized Riemann Hypothesis for the Dedekind zeta function of the cyclotomic field $\mathbb{Q}(e^{2\pi i/q})$ is true if and only if for all integers $k\ge 1$ we have \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,1)}.\] Here ${\bar{N}}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equiv 1(\mathrm{mod}q)$ and $C(q,1)$ is the constant appearing in the asymptotic formula \[\prod_{\substack{p \leq x \\ p \equiv 1~({\rm mod}~{q})}} \left(1 - \frac{1}{p}\right) \sim \frac{C(q, 1)}{(\log{x})^\frac{1}{φ(q)}},\] as $x\to \infty$. We also prove that, for $q\le 400,000$ and integers $a$ coprime to $q$, the analogous inequality \[\frac{\bar{N}_k}{φ(\bar{N}_k)(\log(φ(q)\log{\bar{N}_k}))^{\frac{1}{φ(q)}}} > \frac{1}{C(q,a)}\] holds for infinitely many values of $k$. If in addition $a$ is a not a square modulo $q$, then there are infinitely many $k$ for which this inequality holds and also infinitely many $k$ for which this inequality fails.