On sums of four smooth squares (Q1293129)

The author applies the Hardy-Littlewood circle method together with some of his results in [Acta Arith. 80, 165-185 (1997; Zbl 0871.11066)] to continue his investigation of that work. In the present paper he obtains: Let \(F(M)= (\log M \log \log M)^{1/2}\) and \(P(n)\) denote the greatest prime factor of the integer \(n\). Then almost every natural number \(M\) not divisible by 8 has at least \(M \exp(-F(M)/8)\) representations of the form \[ n_1^2+ n_2^2+ n_3^2+ n_4^2= M, \] where \(n_j\) are integers satisfying \[ P(n_1 n_2 n_3 n_4)< \exp(20 F(M)). \] More precisely, the number of exceptions up to \(N\) is \(\ll N\exp (-F(N)/16)\).