Small values of the Euler function and the Riemann hypothesis

TL;DR

This paper investigates the behavior of a specific function related to Euler's totient and the Riemann hypothesis, showing boundedness under the hypothesis and unboundedness if it fails.

Contribution

It provides a new criterion involving the function c(n) that characterizes the truth of the Riemann hypothesis based on its boundedness properties.

Findings

c(N_k) is bounded under Riemann's hypothesis

c(N_k) is unbounded if Riemann's hypothesis fails

Explicit bounds for c(N_k) are derived

Abstract

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the product of the first $k$ primes. In this article, we consider the function $c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$. Under Riemann's hypothesis, it is proved that $c(N_k)$ is bounded and explicit bounds are given while, if Riemann's hypothesis fails, $c(N_k)$ is not bounded above or below.