UDC 511

MATHEMATICS

A. A. POLYANSKII

ON ONE ANALOGUE OF THE HARDY–LITTLEWOOD EQUATION

(Presented by Academician Yu. V. Linnik on 19 VI 1967)

In the monograph of Yu. V. Linnik (¹), among analogues of the Hardy–Littlewood equation, the equation

\[ p_1p_2+\xi^2+\eta^2=n, \tag{1} \]

is considered, where \(p_1\) and \(p_2\) independently run through the prime numbers; \(\xi\) and \(\eta\) run through the integers; \(n\) is a sufficiently large natural number (the main parameter).

For the number \(S(n)\) of solutions of equation (1), in (¹) an asymptotic formula is derived under the additional restrictions

\[ p_1,\ p_2>\exp(\ln\ln n)^2. \]

The aim of the present note is to remove these restrictions and to obtain \(S(n)\) with a better remainder term than in (¹).

Theorem. As \(n\to\infty\),

\[ S(n)=\pi A_0 A(n)\frac{n}{\ln n} \prod_{p\mid n}\frac{(p-1)(p-\chi(p))}{p^2-p+\chi(p)} +O\left(\frac{n}{(\ln n)^{1.042}}\right), \tag{2} \]

where

\[ A_0=\prod_{p>2}\left(1+\frac{\chi(p)}{p(p-1)}\right); \quad \chi(m)\ \text{is the nonprincipal character mod }4;\quad A(n)\ \text{is} \]

some arithmetical factor, whose structure is clear from the proof.

We precede the proof of the theorem with two lemmas. Consider the equation

\[ ap+b(\xi^2+\eta^2)=n, \tag{3} \]

where \(a,b\) are integers satisfying the conditions: \(a=O(\exp(\ln n)^\alpha)\), \(b=O(\ln^C n)\), \(C>0\) is a sufficiently large constant, and \(p\) runs through the prime numbers. We shall assume that \((a,b)=1\), \((ab,2n)=1\). The latter conditions are not essential.

Let \(Q(n)\) be the number of solutions of equation (3).

Lemma 1. As \(n\to\infty\),

\[ Q(n)=\pi A_0\frac{n}{ab\ln n} \prod_{p\mid an}\frac{(p-1)(p-\chi(p))}{p^2-p+\chi(p)} \prod_{p\mid b}\frac{p^2}{p^2-p+\chi(p)} +O\left(\frac{n}{ab(\ln n)^{1.042}}\right). \tag{4} \]

For the proof of the lemma, following C. Hooley (⁴) and B. M. Bredikhin (⁴), we represent \(Q(n)\) in the following form:

\[ \begin{aligned} Q(n) &=\sum_{ap+b(\xi^2+\eta^2)=n}1 =4\sum_{ap+2^\lambda bxy=n}\chi(x) =8\sum_{\substack{ap+2^\lambda bxy=n\\ x\le \sqrt n\,n_1^{-1}}}\chi(x) \\ &\quad -4\sum_{\substack{ap+2^\lambda bxy=n\\ \sqrt n\,n_1^{-1}<x<\sqrt n\,n_1\\ y<\sqrt n\,n_1}}\chi(x) +4\sum_{\substack{ap+2^\lambda bxy=n\\ \sqrt n\,n_1^{-1}<x<\sqrt n\,n_1}}\chi(x) +O\left(\frac{n}{ab\ln^2 n}\right) \\ &=\Sigma_A-\Sigma_{B_1}+\Sigma_{B_2} +O\left(\frac{n}{ab\ln^2 n}\right), \end{aligned} \]

where \(n_1=\exp(\ln n)^{\alpha_1}\); \(\alpha_0,\alpha_1>0\) are sufficiently small constants.

\(\Sigma_A\) is evaluated with the aid of E. Bombieri’s lemma \((^5)\), while \(\Sigma_{B_1}\) and \(\Sigma_{B_2}\) are estimated by S. Hooley’s method \((^4)\) according to the scheme developed in \((^8)\).

Equation (3) may be regarded as a special case of the linear equation \(ax+by=n\), when \(x\) runs through the primes and \(y\) assumes values of the quadratic form \(\varphi(\xi,\eta)=\xi^2+\eta^2\). This equation is a natural generalization of the classical Hardy–Littlewood equation and is of interest from the point of view of possible applications (see, for example, below and \((^7,^8)\)).

Next, let \(S_1(n)\) be the number of solutions of equation (1) when

\[ p_1,\ p_2>P_0=\exp(\ln n)^{\alpha_0}. \]

Lemma 2. As \(n\to\infty\),

\[ S_1(n)=\pi A_0\prod_{p\mid n}\frac{(p-1)(p-\chi(p))}{p^2-p+\chi(p)} \sum_{\substack{p_1p_2<n\\ p_1p_2>P_0}}1 +O\left(\frac{n}{(\ln n)^{1.042}}\right). \tag{5} \]

Proof. We write \(S_1(n)\) in the form

\[ S_1(n)=2 \sum_{\substack{p_1p_2+\xi^2+\eta^2=n\\ P_0<p_1<P}}1 + \sum_{\substack{p_1p_2+\xi^2+\eta^2=n}}1 +O\left(\frac{n}{\ln^2 n}\right) = \]

\[ =2S_2(n)+S_3(n)+O\left(\frac{n}{\ln^2 n}\right), \tag{6} \]

where \(P=\exp\ln n\,\dfrac{\ln\ln\ln n}{K\ln\ln n}\), and \(K\) is a sufficiently large constant.

With the aid of the dispersion method \((^1)\) (see \((^6,^13,^10)\) and p. 131) we obtain

\[ S_2(n)=\pi A_0\prod_{p\mid n}\frac{(p-1)(p-\chi(p))}{p^2-p+\chi(p)} \sum_{\substack{p_1p_2<n\\ P_0<p_1<P}}1 +O\left(\frac{n}{\ln^2 n}\right). \tag{7} \]

Refining the proof of Theorem 2 of \((^2)\), we obtain

\[ S_3(n)=\pi A_0\prod_{p\mid n}\frac{(p-1)(p-\chi(p))}{p^2-p+\chi(p)} \sum_{\substack{p_1p_2<n\\ p_1,\ p_2>P}}1 +O\left(\frac{n}{(\ln n)^{1.042}}\right). \tag{8} \]

From (6)—(8), (5) follows.

Proof of the theorem. Decompose \(S(n)\) as follows:

\[ S(n)=2 \sum_{\substack{p_1p_2+\xi^2+\eta^2=n\\ p_1<P_0}}1 +S_1(n)+O\left(\frac{n}{\ln^2 n}\right) = 2S_4(n)+S_1(n)+O\left(\frac{n}{\ln^2 n}\right). \tag{9} \]

We find \(S_4(n)\):

\[ S_4(n)= \sum_{\substack{(p_1,n)=1\\ p_1<P_0}} \sum_{\substack{p_1p_2+\xi^2+\eta^2=n}}1 + \sum_{\substack{p_1\mid n\\ p_1<P_0}} \sum_{\substack{p_1p_2+\xi^2+\eta^2=n}}1 =\Sigma_1+\Sigma_2. \tag{10} \]

The inner sum in \(\Sigma_1\) is evaluated with the aid of Lemma 1. We obtain

\[ \Sigma_1= \sum_{\substack{(p_1,n)=1\\ p_1<P_0}} \left( \pi A_0\frac{n}{p_1\ln n} \prod_{p\mid p_1n}\frac{(p-1)(p-\chi(p))}{p^2-p+\chi(p)} + O\left(\frac{n}{p_1(\ln n)^{1.042}}\right) \right). \tag{11} \]

Further, by virtue of

\[ \sum_{\substack{d\mid ml\\ (m,l)=1}}\chi(d) = \sum_{d\mid m}\chi(d)\sum_{d\mid l}\chi(d) \]

and Lemma 1, we have (with ad-

with admissible error)

\[ \begin{aligned} \Sigma_2 &=4 \sum_{\substack{p_1\mid n\\ p_1<P_0}} \sum_{\lambda,\ s=0,1\ldots} \sum_{d\mid p_1^{1+s}} \chi(d) \sum_{p_2+2^\lambda p_1^sxy=n_0}\chi(x) \\ &= \sum_{\substack{p_1\mid n\\ p_1<P_0}} \sum_{s=0,1,\ldots} \sum_{d\mid p_1^{1+s}} \chi(d) \left( \pi A_0 \frac{n}{p_1^{1+s}\ln n} \prod_{p\mid n_0} \frac{(p-1)(p-\chi(p))}{p^2-p+\chi(p)} \frac{p_1^2}{p_1^2-p_1+\chi(p_1)} +\right. \\ &\hspace{8em}\left. +O\left(\frac{n}{p_1^{1+s}(\ln n)^{1.042}}\right) \right), \end{aligned} \tag{12} \]

where \(n_0=n/p_1\).

Now (2) follows from (5), (9)–(12).

By the methods of the papers \((^6,^8,^9)\) one can derive geometric and ergodic properties of the solutions of equation (1).

Equation (1) can be generalized. By applying analogous means one can treat the equation

\[ p_1p_2\ldots p_k+\varphi(\xi,\eta)=n, \]

where \(k>2\), and \(\varphi(\xi,\eta)\) is some prescribed quadratic form.

Kuibyshev Pedagogical Institute
named after V. V. Kuibyshev

Received
13 VI 1967

REFERENCES

\(^1\) Yu. V. Linnik, The dispersion method in binary additive problems, L., 1961.
\(^2\) Yu. V. Linnik, Matem. sborn., 52 (94), 2, 661 (1960).
\(^3\) B. M. Bredikhin, UMN, 20, No. 2, 89 (1965).
\(^4\) C. Hooley, Acta Math., 97, 189 (1957).
\(^5\) E. Bombieri, On the Large Sieve, Milano, 1965.
\(^6\) A. A. Polyanskii, DAN, 168, No. 1 (1966).
\(^7\) A. A. Polyanskii, The solution of certain binary equations of Hardy–Littlewood type, Dissertation, Kuibyshev, 1966.
\(^8\) B. M. Bredikhin, Yu. V. Linnik, DAN, 166, No. 6, 1267 (1966).
\(^9\) A. I. Vinogradov, Matem. zametki, 1, No. 2, 189 (1967).