On the distribution of squares of integers in rational quaternion algebrs (Q1858611)

Let \(\gamma\) be a rational integer and \(j\) a hypercomplex number with \(j^2= \gamma\). Consider the set \(Q\langle \gamma\rangle= Q[i]+ Q[i]j\) and define addition and multiplication such that \(zj= j\overline{z}\) for all \(z\in Q[i]\). Take \(\gamma\) in such a way that the quaternion algebra \(Q\langle \gamma\rangle\) is a division algebra. The paper is concerned with the functions \[ A_k(\gamma;X)= \#\{(a+bj)^2\in M_k(X)\mid a,b\in \mathbb{Z}[i]\} \quad (k=1,2), \] where \[ \begin{aligned} M_1(X) &= \{(\alpha+\beta j)^2\in Q\langle \gamma\rangle\mid \alpha,\beta\in \mathbb{C},\;|\operatorname {Re}(\alpha)|, |\operatorname {Im}(\alpha)|, |\operatorname {Re}(\beta)|, |\operatorname {Im}(\beta)|\leq X\},\\ M_2(X) &= \{(\alpha+ \beta j)^2\in Q\langle \gamma\rangle\mid \alpha,\beta\in \mathbb{C},\;|\alpha|, |\beta|\leq X\}. \end{aligned} \] The author generalizes his own results concerning Hamilton's quaternions \((\gamma=-1)\) [Acta Arith. 93, 359-372 (2000; Zbl 0947.11028), ibid. 101, 81-95 (2002; Zbl 1004.11054)] and proves asymptotic representations for \(A_k (\gamma;X)\), where the error term has the same quality as in Dirichlet's divisor problem.