On the Distribution of Squares of Hypercomplex Integers

Abstract

Let A be a real quadratic algebra of dimension s⩾3 which satisfies the basic relations of hypercomplex systems. For a large positive parameter X, let A(X) denote the number of squares α2 with α∈A, α integral, and all s components of α2 lying in the interval [−X, X]. With particular regard to Cayley's octaves, and generalizing former results concerning Gaussian integers by H. Müller and W. G. Nowak, and Hurwitz integral quaternions by the author, we show thatA(X)=cXs/2−dX(s−1)/2+O(X(logX)−1/2+X(s−2)/2δ(X))(X→∞),where c and d are certain positive constants depending on s, and δ(X) is any upper bound of the error term in the divisor problem, e.g. δ(X)=X23/73+ε.