Gauss circle problem over smooth integers

For a positive integer , let 2( ) be the number of representations of as sums of two squares (of integers), where the convention is that different signs and different orders of the summands yield distinct representations. A famous result of Gauss shows that ( ) ∶= ∑ ≤ 2( ) ∼ . Let ( ) denote the largest prime factor of and let ( , ) ∶= { ≤ ∶ ( ) ≤ }. In this paper, we study the asymptotic behavior of ( , ) ∶= ∑ ∈ ( , ) 2( ) for various ranges of 2 ≤ ≤ . For in a certain large range, we show that ( , ) ∼ ( ) ⋅ where ( ) is the Dickman function and = log ∕ log . We also obtain the asymptotic behavior of the partial sum of a generalized representation function following a method of Selberg.