A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous Waring equations
The author proves the solubility of the equation \(n=x^2+y^2+6pz^2\), where \(p\) is a fixed large prime, under some natural congruential conditions and the assumption \(4nm^{12}>p^{21+\varepsilon}\), where \(n=tm^2, (t,m)=1\). This enables her to establish the solubility of the equation \(n=x^2+y^2+z^3+u^3+v^4+w^{14}+t^{4k+1}\) for all sufficiently large \(n\) and \(k\), improving an earlier result of her own. Note that the sum of the reciprocals of the exponents is less than 2 once \(k\) is sufficiently large. The proof of these results uses a modification of methods of \textit{Yu. V. Linnik} [Acta Arith. 21, 413--422 (1972; Zbl 0247.10014)], appeals to results of \textit{P. X. Gallagher} [Invent. Math. 16, 191--201 (1972; Zbl 0246.10030)], and involves estimates for Fourier coefficients of cusp forms of weight \(3/2\) due to \textit{H. Iwaniec} [Invent. Math. 87, 385--401 (1987; Zbl 0606.10017)] and \textit{W. Duke, R. Schulze-Pillot}, [Invent. Math. 99, No. 1, 49--57 (1990; Zbl 0692.10020)].
scientific article; zbMATH DE number 5659049