Distribution functions of the sequence phi(n)/n, n in (k,k+N]

Vladimir Balaz (1) , Pierre Liardet (2) , Oto Strauch (3)

1 Department of Mathematics
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités
3 Mathematical Institute of the Slovak Academy of Sciences

It is well known that the sequence $\varphi(n)/n$, n=1,2,... has a singular asymptotic distribution function. P. Erdös in 1946 found a sufficient condition on the sequence of intervals (k,k+N], such that phi(n)/n, n in (k,k+N], has the same singular function. In this note we prove a sufficient and necessary condition. For simplifying the necessary condition we express the sum \sum_{k n*k+N(!(n) ¡ log logN)2, where !(n) is the number of di®erent primes divided n.

Mots clés: Euler function; sequence; distribution function

Domaines: Théorie des nombres [math.NT]

Vladimir Balaz, Pierre Liardet, Oto Strauch. Distribution functions of the sequence phi(n)/n, n in (k,k+N]. Integers : Electronic Journal of Combinatorial Number Theory, 2010, 10, pp.705--732. ⟨hal-00871083⟩