Values of the Euler and Carmichael Functions Which Are Sums of Three Squares

Let φ denote Euler’s totient function. The frequency with which φ(n) is a perfect square has been investigated by Banks, Friedlander, Pomerance, and Shparlinski, while the frequency with which φ(n) is a sum of two squares has been studied by Banks, Luca, Saidak, and Shparlinski. Here we look at the corresponding threesquares question. We show that φ(n) is a sum of three squares precisely seveneighths of the time. We also investigate the analogous problem with φ replaced by Carmichael’s λ-function. We prove that the set of n for which λ(n) is a sum of three squares has lower density > 0 and upper density < 1.