On the distribution of squares of integral quaternions. II (Q2773320)

Let \(q= a+ib+jc+kd\), where \(a,b,c,d\in \mathbb{Z}\) or \(a,b,c,d\in \mathbb{Z}+ \frac 12\), run over integral quaternions, and write \(q^2= \alpha+ i\beta+ j\gamma+k\delta\). It is shown that NEWLINE\[NEWLINE\#\{q^2: \alpha^2+ \beta^2\leq X^2,\;\gamma^2+ \delta^2\leq X^2\}= C_1X^2+ C_2X^{3/2}+ O(X^{96/73} (\log x)^{461/146})NEWLINE\]NEWLINE for certain constants \(C_1\), \(C_2\). In the previous paper of this series [Acta Arith. 93, 359-372 (2000; Zbl 0947.11028)] the author estimated similarly the number of squares \(q^2\) with \(|\alpha|, |\beta|, |\gamma|, |\delta|\leq X\), and also those for which \(|\alpha|\leq X\) and \(\beta^2+ \gamma^2+ \delta^2\leq x^2\). The proof of the present estimate uses \textit{M. N. Huxley}'s results [Area, lattice points and exponential sums. Oxford Univ. Press (1996; Zbl 0861.11002)] on the circle problem, along with other bounds derived from exponential sums.