Euler’s function on products of primes in a fixed arithmetic progression

Abstract:

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler’s function $\varphi (n)$ and the Riemann Hypothesis. Let $C(q, a)$ be the constant appearing in the asymptotic formula \[ \prod _{\substack {p \leq x \\ p \equiv a\ \text {(mod\ {q})}}} \left (1 - \frac {1}{p}\right ) \sim \frac {C(q, a)}{(\log {x})^\frac {1}{\varphi (q)}},\] as $x\rightarrow \infty$. Among other things, we prove that for $1\leq q\leq 10$ 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 $k\geq 1$ we have \[ \frac {\overline {N}_k}{\varphi (\overline {N}_k)(\log (\varphi (q)\log {\overline {N}_k}))^{\frac {1}{\varphi (q)}}} > \frac {1}{C(q,1)}.\] Here $\overline {N}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equiv 1\ \text {(mod\ {q})}$. We also prove that, for $q\leq 400,000$ and integers $a$ coprime to $q$, the analogous inequality \[ \frac {\overline {N}_k}{\varphi (\overline {N}_k)(\log (\varphi (q)\log {\overline {N}_k}))^{\frac {1}{\varphi (q)}}} > \frac {1}{C(q,a)}\] holds for infinitely many values of $k$. If in addition $a$ is 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.