Small values of the Euler function and the Riemann hypothesis

Jean-Louis Nicolas (1)

1 CTN - Combinatoire, théorie des nombres

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.

Mots clés: Euler's function; Riemann hypothesis; Explicit formula

Domaines: Théorie des nombres [math.NT]

Jean-Louis Nicolas. Small values of the Euler function and the Riemann hypothesis. Acta Arithmetica, 2012, 155 (3), pp.311-321. ⟨10.4064/aa155-3-7⟩. ⟨hal-00666154⟩