Euler's Function on Products of Primes in Progressions

TL;DR

This paper explores the relationship between Euler's totient function on products of primes in arithmetic progressions and the Riemann Hypothesis, establishing equivalences and infinite occurrence results for certain inequalities.

Contribution

It generalizes results relating small values of Euler's function to the Riemann Hypothesis for cyclotomic fields and proves infinite occurrence of specific inequalities for primes in progressions.

Findings

Equivalence between Riemann Hypothesis for certain cyclotomic fields and inequalities involving prime products.

Proved that for large q, the inequalities hold infinitely often for primes in arithmetic progressions.

Established conditions under which inequalities both hold and fail infinitely often depending on quadratic residues.

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. 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 \[\frac{\bar{N}_k}{\varphi(\bar{N}_k)(\log(\varphi(q)\log{\bar{N}_k}))^{\frac{1}{\varphi(q)}}} > \frac{1}{C(q,1)}.\] Here $\bar{N}_k$ is the product of the first $k$ primes in the arithmetic progression $p\equiv 1~({\rm mod}~{q})$ and $C(q, 1)$ is the constant appearing in the asymptotic formula \[\prod_{\substack{p \leq x \\ p \equiv 1~({\rm mod}~{q})}} \left(1 - \frac{1}{p}\right) \sim \frac{C(q, 1)}{(\log{x})^\frac{1}{\varphi(q)}},\] as $x\rightarrow\infty$. We also prove that, for $q\leq 400,000$ and integers $a$ coprime to $q$, the analogous inequality \[\frac{\bar{N}_k}{\varphi(\bar{N}_k)(\log(\varphi(q)\log{\bar{N}_k}))^{\frac{1}{\varphi(q)}}} > \frac{1}{C(q,a)}\] 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.