Euler's Function on Products of Primes in Progressions

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 $\mathbb{Q}(e^{2\pi i\mathrm{/}q})$ is true if and only if for all integers $k\ge 1$ we have [\frac{\bar{N}_k}{\varphi(\bar{N}_k)(\log(\varphi(q)\log{\bar{N}_k}))^{\frac{1}{\varphi(q)}}}

\frac{1}{C(q,1)}.] Here ${\bar{N}}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}q)$ and $C(q,1)$ is the constant appearing in the asymptotic formula $$\prod _{\begin{array}{c}p\le x\\ p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}q)\end{array}}(1-\frac{1}{p})\sim \frac{C(q,1)}{(\log x{)}^{\frac{1}{\phi (q)}}},$$ as $x\to \mathrm{\infty }$. We also prove that, for $q\le 400,000$ and integers $a$ coprime to $q$, the analogous inequality $$\frac{{\bar{N}}_k}{\phi ({\bar{N}}_k)(\log (\phi (q)\log {\bar{N}}_k){)}^{\frac{1}{\phi (q)}}}\frac{1}{C(q,a)}$$ 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.