Square-free values of the Carmichael function.
For a positive integer \(n\) the Carmichael function \(\lambda(n)\) is the exponent of the multiplicative group modulo \(n\). In this paper, the authors find an asymptotic formula for the number \(L(x)\), which counts, for a given positive real number \(x\), the number of positive integers \(n\leq x\) having the property that \(\lambda(n)\) is square-free. Their result is that \[ L(x)\approx \kappa {{x}\over {\log^{1-\alpha} x}},\tag{1} \] where \(\alpha=\prod_{p\geq 2} (1-1/(p(p-1)))\) is the Artin constant and \(\kappa\) is a more complicated constant (explicitly determined). For the proof, the authors first use the Bombieri-Vinogradov theorem to show that if \(\pi_{sf}(x)\) counts the number of primes \(p\leq x\) with \(p-1\) square-free, then \(\pi_{sf}(x)=\alpha\pi(x)+O(x/\log^A x)\), where \(A>0\) is an arbitrary constant. Once equipped with this fact, the main result follows from the observation that the characteristic function of the set of those \(n\) such that \(\lambda(n)\) is square-free is multiplicative and supported on \(\pi_{sf}\), together with a well-known result of Wirsing relating the average value of a ``small multiplicative function'' \(f(n)\) to \(\sum_{p\leq x} f(p)\). The paper contains also some other interesting facts, such as a uniform upper bound for the function \(L\) in short intervals, a formula for the counting function of those positive integers \(n\) whose Carmichael function is \(k\)-free (for a fixed integer \(k\geq 2\)), as well as the results of some computations analyzing the speed of convergence to \(1\) of the ratio between \(L(x)\) and the right hand side of (1).
scientific article; zbMATH DE number 2012767
scientific article