Sums of smooth squares

Abstract

AbstractLetR(n,θ) denote the number of representations of the natural numbernas the sum of four squares, each composed only with primes not exceedingnθ/2. Whenθ>e−1/3 a lower bound forR(n,θ) of the expected order of magnitude is established, and whenθ>365/592, it is shown thatR(n,θ)>0 holds for largen. A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixedθ>0. This last result is the key to boundR(n,θ) from below.