Mathematics

V. P. Myakishev

DISTRIBUTION OF PRIMITIVE INTEGER POINTS ON CERTAIN CONES

(Presented by Academician I. M. Vinogradov on 22 XI 1961)

Let there be a surface of second order

\[ f(x,y,z)=0. \]

We assume that the coefficients of the quadratic form \(f(x,y,z)\) are rational integers. A solution \((x,y,z)\) of this equation in rational integers satisfying \((x,y,z)=1\) will be called a primitive integer point. \((\ )\) denotes the greatest common divisor.

Theorem 1. For the number of primitive integer points \(A(h)\) lying on the surface of the cone \(x^2+y^2-z^2=0\) in the region \(0<z\le h\), the following asymptotic formula holds: as \(h\to\infty\),

\[ A(h)=\frac{4}{\pi}h+O(\sqrt h). \]

Proof. The problem of the number of primitive integer points \(A(h)\) lying on the surface of the cone \(x^2+y^2-z^2=0\) in the region \(0<z\le h\) we reduce to the problem of primitive integer solutions of the equation

\[ x^2+y^2=z^2. \]

The following lemmas are known (see, for example, (1), problem 9a, pp. 23, 116, and question 176, chap. II, pp. 36, 127):

Lemma 1. All primitive integer solutions of the equation \(x^2+y^2=z^2\) with \(x\ge0,\ y\ge0,\ z>0\) are obtained uniquely from the formulas:

\[ \begin{cases} x=2mn,\\ y=m^2-n^2,\\ z=m^2+n^2, \end{cases} \qquad m>0,\ n\ge0,\ m>n,\ (m,n)=1, \]
\[ m \text{ and } n \text{ of different parity} \]

\[ \begin{cases} x=m^2-n^2,\\ y=2mn,\\ z=m^2+n^2, \end{cases} \qquad m>0,\ n\ge0,\ m>n,\ (m,n)=1, \]
\[ m \text{ and } n \text{ of different parity}. \]

Lemma 2. Let \(k>1\) and suppose systems are given

\[ x_1', x_2', \ldots, x_k';\quad x_1'', x_2'', \ldots, x_k'';\ \ldots;\quad x_1^{(n)}, x_2^{(n)}, \ldots, x_k^{(n)}, \]

each of which consists of integers not all simultaneously zero. Suppose, further, that for these systems a function \(f(x_1,x_2,\ldots,x_k)\) is uniquely defined. Then

\[ S'=\sum \mu(d) S_d, \]

where \(S'\) denotes the sum of the values \(f(x_1,x_2,\ldots,x_k)\) extended over systems of relatively prime numbers, and \(S_d\) denotes the sum of the values \(f(x_1,x_2,\ldots,x_k)\) extended over systems of numbers simultaneously divisible by \(d\). Here \(d\) runs through the positive integers.

From Lemma 1 it follows that

\[ A(h)=4M(h)-4, \]

where \(M(h)\) is the number of integer points lying inside the quarter circle \(m^2+n^2\le h,\ m\ge 0,\ n\ge 0\), with \((m,n)=1\), and with \(m\) and \(n\) of different parity.

Finally, using Lemma 2, we reduce the problem of the number \(M(h)\) of integer points with the indicated conditions to the problem of the number of integer points inside a circle. Applying the known estimate for the number \(K(R)\) of integer points inside the circle \(x^2+y^2\le R\),

\[ K(R)=\pi R+O(R^{1/3}), \]

we obtain the required result.

Fig. 1

By means of the method used in the proof of Theorem 1, one can prove a theorem on the distribution of primitive integer points inside a sector of aperture \(\varphi\) lying on the surface of the cone \(x^2+y^2-z^2=0\) (Fig. 1).

Theorem 2. Let \(A(h,\varphi)\) be the number of primitive integer points lying on the surface of the cone \(x^2+y^2-z^2=0\) in the region under consideration. Then, as \(h\to\infty\),

\[ A(h,\varphi)=\frac{2\theta}{\pi^2}h+O(\sqrt h\ln h), \]

where \(\varphi\) and \(\theta\) are connected by the relation

\[ 2\cos\varphi-\cos\theta=1. \]

Theorem 1 can be generalized. Denote by \(F(h)\) the number of primitive integer points lying on the surface of the cone

\[ u^2+Av^2=w^2 \]

in the region \(0<w\le h\), where \(A\) is a positive integer not divisible by the square of any integer other than 1.

Theorem 3. As \(h\to\infty\),

\[ F(h)= \begin{cases} \dfrac{2^{\,n+2}\sqrt A}{\pi\displaystyle\prod_{i=1}^{n}(A_i+1)}\,h +O(\sqrt h\ln h), & \text{if } A=\displaystyle\prod_{i=1}^{n}A_i,\ A_i\ne 1,2; \\[1.2em] \dfrac{2^{\,m+1}\sqrt A}{\pi\displaystyle\prod_{i=1}^{m}(A'_i+1)}\,h +O(\sqrt h\ln h), & \text{if } A=2\displaystyle\prod_{i=1}^{m}A'_i,\ A'_i\ne 1,2; \\[1.2em] \dfrac{2\sqrt2}{\pi}\,h+O(\sqrt h\ln h), & \text{if } A=2. \end{cases} \]

I express my gratitude to Academician I. M. Vinogradov for his help in this work.

Received
17 XI 1961

REFERENCES CITED

1. I. M. Vinogradov, Fundamentals of Number Theory, Moscow, 1952.