The representation function for sums of three squares along arithmetic progressions

For positive integers n, let r(n) = #{(x, y, z)[member of] [Z.sup.3] : [x.sup.2] + [y.sup.2] + [z.sup.2] = n}. Let g be a positive integer, and let A mod M be any congruence class containing a squarefree integer. We show that there are infinitely many squarefree positive integers n [equivalent to] A mod M for which g divides r(n). This generalizes a result of Cho. Key words: Class number; imaginary quadratic field; three squares.