Representation of Integers by Positive Quaternary Quadratic Forms

Let $f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2$ , where $D >1$ is an integer such that $D ne d^2$ and ${{sqrt n } mathord{left/ {vphantom {{sqrt n } {sqrt D = n^theta , 0 < theta < {1 mathord{left/ {vphantom {1 2}} right. kern-0em} 2}}}} right. kern-0em} {sqrt D = n^theta , 0 < theta < {1 mathord{left/ {vphantom {1 2}} right. kern-0em} 2}}}$ . Let $rf(n)$ be the number of representations of n by f. It is proved that $r_f (n) = pi ^2 frac{n}{{sqrt D }}sigma _f (n) + Oleft( {frac{{n^{1 + varepsilon - c(theta )} }}{{sqrt D }}} right),$ where $sigma _f (n)$ is the singular series, $c(theta ) >0$ , and ε is an arbitrarily small positive constant. Bibliography: 14 titles.