Fomenko, O. M. Representation of integers by positive quaternary quadratic forms. (English. Russian original) Zbl 1130.11314 J. Math. Sci., New York 118, No. 1, 4904-4909 (2003); translation from Zap. Nauchn. Semin. POMI 276, 291-299 (2001). Summary: Let \(f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2\), where \(D >1\) is an integer such that \(D \neq d^2\) and \({\sqrt n }/ {\sqrt D} = n^\theta\), \(0 < \theta < \frac12\). Let \(r_f(n)\) be the number of representations of \(n\) by \(f\). It is proved that\[ r_f (n) = \pi ^2 \frac{n}{{\sqrt D }}\sigma _f (n) + O\left( {\frac{{n^{1 + \varepsilon - c(\theta )} }}{{\sqrt D }}} \right), \]where \(\sigma _f (n)\) is the singular series, \(c(\theta ) >0\), and \(\varepsilon\) is an arbitrarily small positive constant. MSC: 11E25 Sums of squares and representations by other particular quadratic forms 11E20 General ternary and quaternary quadratic forms; forms of more than two variables × Cite Format Result Cite Review PDF Full Text: DOI