UDC 511

MATHEMATICS

E. P. GOLUBEVA

ON THE REPRESENTATION OF LARGE NUMBERS BY TERNARY QUADRATIC FORMS

(Presented by Academician I. M. Vinogradov on 30 VII 1969)

Let (f(x_1,x_2,x_3)) be a nondegenerate integral quadratic form. It is known (\left({}^{1,2}\right)) that the question of the asymptotic distribution of integer points on the surface

[
f(x_1,x_2,x_3)=n
\tag{1}
]

as (n\to\infty) is closely connected with the question of the boundary of the zeros of Dirichlet (L)-functions with real characters. In this note we present conditional theorems on the uniform distribution of integer points on the surface (1) in the case when (f) is a diagonal form.

Throughout what follows, unless the contrary is stated, the following is assumed to hold.

Assumption. There exists a number (\varepsilon>0) such that, for sufficiently large integral (|m|), the function

[
L(s,\chi)=\sum_{d=1}^{\infty}\frac{\left(\frac{m}{d}\right)}{d^s}\qquad(\operatorname{Re}s>1)
]

has no zeros in the rectangle

[
1-\frac{1}{(\ln |m|)^{1-\varepsilon}}<\operatorname{Re}s\leqslant 1,\qquad |\operatorname{Im}s|\leqslant 1.
]

We shall use the following notation: (f(x_1,x_2,x_3)) is a diagonal integral quadratic form of determinant (D_f), whose coefficients are pairwise relatively prime and at least one of them is positive; (n>0) is an integer relatively prime to (D_f) and such that the congruence

[
f(x_1,x_2,x_3)\equiv n\pmod 8
]

is primitively solvable; the function (L(s,\chi)) is defined for (\operatorname{Re}s>1) by the series

[
L(s,\chi)=\sum_{d=1}^{\infty}\frac{\left(\frac{-nD_f}{d}\right)}{d^s};
]

(\Omega) is an arbitrary finite convex domain on the surface

[
f(x_1,x_2,x_3)=1,
\tag{2}
]

(Q(n,\Omega)) is the number of integer points in the projection of (\Omega) onto the surface with equation (1), (\omega(\Omega)) is the volume of the cone with base (\Omega) and vertex at (O); if (f) is a positive form, then (Q_f(n)) is the number of integer points on the ellipsoid (1); (\eta_0) is a positive constant depending only on (D_f) and (\varepsilon).

Theorem 1. Let (f(x_1,x_2,x_3)) be a positive form; then

[
Q(n,\Omega)=c_0\omega(\Omega)Q_f(n)+O\left(\frac{n^{1/2}L(1,\chi)}{(\ln n)^{\gamma_{10}}}\right);
]

(c_0) and the constant entering into the remainder do not depend on (\Omega).

The method of proof of this theorem is analogous to the method proposed by Yu. V. Linnik (see ((^1)), p. 189) for studying the distribution of integral points on the sphere by segments; however, instead of ergodic considerations we use the scheme of work ((^3)).

Yu. V. Linnik ((^4)) and A. V. Malyshev (see ((^2)), Ch. V), with the aid of the arithmetic of quaternions, proved a number of theorems on the representation of numbers by a given positive ternary quadratic form. From A. V. Malyshev’s results it follows, in particular, that for a form (f) with odd coefficients, for some constant (\varkappa>0),

[
Q_f(n)>\varkappa n^{1/2}L(1,\chi),
]

if (L(s,\chi)) has no zeros in the circle

[
|s-1|<\frac{(\ln\ln n)^2\ln\ln\ln n}{(\ln n)^{1/2}}.
]

Asymptotic formulas for the number of solutions of equation (1) are known only for a comparatively narrow class of positive forms (see ((^2)), Ch. VI). Here we indicate one more class of forms for which one can find an asymptotic formula for (Q_f(n)).

Theorem 2. Let (f(x_1,x_2,x_3)) be a positive form such that, if (p) and (q) are arbitrary prime divisors of (D_f), then (p\equiv 1(\operatorname{mod}4)) and ((p/q)=1); then

[
Q_f(n)=\prod_{p/D_f}\frac{1}{p+1}\,Q(nD_f)+O\left(\frac{Q(nD_f)}{(\ln n)^{\gamma_{10}}}\right),
]

where (Q(nD_f)) is the number of integral points on the sphere (x^2+y^2+z^2=nD_f).

In the works of Yu. V. Linnik ((^5)) and B. F. Skubenko ((^6)), the asymptotic-geometric properties of integral points on the surface (1) were studied in the two most important cases of the indefinite form (f), when (D_f=\pm1), connected with the theory of reduction of binary quadratic forms. The distribution of integral points on hyperboloids of general type has not hitherto been considered.

Theorem 3. Let (f(x_1,x_2,x_3)) be an indefinite form,

[
f(x_1,x_2,x_3)=a_1x_1^2+a_2x_2^2-a_3x_3^2,
]

where (a_2,a_3>0) and (|D_f|) is not a perfect square; suppose further that (\Omega_0,\Omega) are two regions on the hyperboloid (2), not containing points of the form ((x_1,x_2,0)), if (a_1>0), and of the form ((0,x_2,0)) in the opposite case; then

[
Q(nD)=\frac{\omega(\Omega)}{\omega(\Omega_0)}Q(n,\Omega_0)+O\left(\frac{n^{1/2}L(1,\chi)}{(\ln n)^{\gamma_{10}}}\right).
]

The proof of this theorem differs little from the proof of Theorem 1.

Theorem 4. Let

[
f(x_1,x_2,x_3)=x_1^2-x_2^2+Dx_3^2,
]

where (D\ne0) is an arbitrary integer; (\Omega_0) is the region on the hyperboloid (2) which is defined by the inequalities

[
1\leq x_1-x_2<2,
]

then

[
2(x_1-x_2)\leqslant x_3<4(x_1-x_2);
]

where

[
Q(n,\Omega_0)=n^{1/2}g(n)\left(1+O\left(\frac{1}{n^{\xi_0}}\right)\right),
]

where

[
g(n)\geqslant cL(1,\chi),
]

(\xi_0) and (c) are some positive constants.

This theorem is unconditional. Its proof relies on the technique of the work ({}^{7}), with the application of results of Burgess ({}^{8}).

In conclusion I express my gratitude to A. I. Vinogradov for his attention to this work and for his advice.

Steklov Mathematical Institute
Academy of Sciences of the USSR
Moscow

Received
30 VII 1969

References

({}^{1}) Yu. V. Linnik, Ergodic Properties of Algebraic Fields, Leningrad, 1967.

({}^{2}) A. V. Malyshev, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 45, 1 (1962).

({}^{3}) A. I. Vinogradov, Mat. Zametki, 1, No. 2, 189 (1967).

({}^{4}) Yu. V. Linnik, Izv. AN SSSR, Ser. Mat., 4, No. 4/5, 363 (1940).

({}^{5}) Yu. V. Linnik, Vestn. LGU, No. 2, 3 (1955); No. 5, 3 (1955); No. 8, 15 (1955).

({}^{6}) B. F. Skubenko, Izv. AN SSSR, Ser. Mat., 26, No. 5, 721 (1962).

({}^{7}) C. Hooley, Math. Zs., 69, No. 3, 15 (1958).

({}^{8}) D. A. Burgess, Proc. Lond. Math. Soc., 12, No. 46, 193 (1962).