Let ? denote Euler’s totient function. The frequency with which ?(n) is a perfect square has been investigated by Banks, Friedlander, Pomerance, and Shparlinski, while the frequency with which ?(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 ?(n) is a sum of three squares precisely seveneighths of the time. We also investigate the analogous problem with ? 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.
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