Quote:
Originally Posted by CRGreathouse
Is there a good upper bound on the number of representations? Even a weak one like r_3(n) < n would be nice.

This should be a common application of sieve theory. Have you tried Halberstam and Richert's text? If that doesn't work, then I would take Bob's advice on contacting Vaughan (although he may be too busy to respond).
I should note that the methods used to achieve an upper bound on the number of representations as the sum of three
squares of primes can be modified without much difficulty to provide a corresponding estimate (of the same quality, at worst) for the sum of three squares. The reason for this is that the situation becomes harder if we restrict some variables to primes. Thus, I am sure that the problem of the sum of three squares of primes (or a variant of this problem, at best) is treated in the HR text.