Divisors of the Euler and Carmichael functions Kevin Ford

Two of the most studied functions in the theory of numbers are Euler’s totient function φ(n) and Carmichael’s function λ(n), the first giving the order of the group (Z/nZ) of reduced residues modulo n, and the latter giving the maximum order of any element of (Z/nZ). The distribution of φ(n) and λ(n) has been investigated from a variety of perspectives. In particular, many interesting properties of these functions require knowledge of the distribution of prime factors of φ(n) and λ(n), e.g., [3], [5], [4], [6], [7], [12], [19]. The distribution of all of the divisors of φ(n) and λ(n) has thus far received little attention, perhaps due to the complicated way in which prime factors interact to form divisors. From results about the normal number of prime factors of φ(n) and λ(n) [5], one deduces immediately that τ(φ(n)) and τ(λ(n)) are each exp{ log 2 2 (log logn)2} for almost all n. However, the determination of the average size of τ(φ(n)) and of τ(λ(n)) is more complex, and has been studied recently by Luca and Pomerance [13]. In this note we investigate problems about localization of divisors of φ(n) and λ(n). Our results have application to the structure of (Z/nZ), since the set of divisors of λ(n) is precisely the set of orders of elements of (Z/nZ). We say that a positive integer m is u-dense if whenever 1 ≤ y < m, there is a divisor of m in the interval (y, uy]. The distribution of u-dense numbers for general u has been investigated by Tenenbaum ([17], [18]) and Saias ([14], [15]). According to Théorème 1 of [14], the number of u-dense integers m ≤ x is ≍ (x log u)/ log x, uniformly for 2 ≤ u ≤ x. In particular, the number of 2-dense integers m ≤ x is ≍ x/ log x, that is, the 2-dense integers are about as sparse as the primes. By contrast, we show that 2-dense values of φ(n) and λ(n) are very common.