MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR Yu. V. LINNIK

SOLUTION OF CERTAIN BINARY ADDITIVE PROBLEMS BY COMPUTING THE VARIANCE IN PROGRESSIONS

In my preceding note \((^{1})\) the concept of the variance of quadratic forms, including the simplest forms \(\varphi=\xi^2+\eta^2\) and \(\varphi=\xi\eta\), in arithmetic progressions was considered, and it was explained how, by means of this concept, one may solve certain binary problems. A further improvement of this method has made it possible to solve binary problems of the form:

\[ n=uv+Q(\xi,\eta), \tag{1} \]

where \(u\) and \(v\) independently run through quite arbitrary sequences of numbers, and \(Q(\xi,\eta)=a\xi^2+b\xi\eta+c\eta^2\) is a binary quadratic form. In this case the corresponding asymptotic laws are obtained.

As an example of the theorems obtained, let us consider the equation

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

where \(p_1\) and \(p_2\) are prime numbers.

Theorem. Let \(\alpha>0\) be a sufficiently small number; \(N_1=n^{1-\alpha}\), \(N_2=n^\alpha\); \(p_1\) runs through primes \(\leq N_1\); \(p_2\) runs through primes \(\leq N_2\); \(\xi^2+\eta^2\) runs through the integers free of squares. Then the number \(H(n)\) of solutions of (2) has the form

\[ \begin{aligned} H(n)\operatorname{Li}(N_1)\operatorname{Li}(N_2)\,\pi &\prod_p\left(1+\frac{\chi_4(p)}{p(p-1)}\right) \prod_{p>2}\left(1-\frac{2+\chi_4(p)-\frac{1}{p}-\frac{\chi_4(p)}{p}}{p^2-p+\chi_4(p)}\right) \\ &\times \prod_{\substack{p\mid n,\,p>2}} \frac{(p-\chi_4(p))(p-1)} {p^2-p-2+\frac{1}{p}+\frac{\chi_4(p)}{p}} + O\!\left(\frac{N_1N_2}{(\ln n)^C}\right), \end{aligned} \tag{3} \]

where \(C\) is an arbitrarily large constant.

Let us briefly explain the ideas of the proof of this theorem. Let

\[ U(m)=\sum_{m=\xi^2+\eta^2}1;\qquad D_1=N_1=n^{1-\alpha};\quad D_2=n^{1-\alpha-\zeta_0} \]

(in what follows \(\zeta_i,\alpha_i\) are small positive constants); \(D\) runs through the numbers \(D_1-D_2\leq D\leq D_1\); \(v\) runs through the prime numbers of the segment \([1,N_2]\) \((N_2=n^\alpha)\).

Let \((QFR)\) be the sequence of integers free of squares. For brevity of exposition (and only for this purpose) we shall henceforth suppose \(u\) to be a prime number.

\[ r(n,D)= \sum_{\substack{m\leq n\\ m=n-Dv\in(QFR)\\ v\ \text{prime}}} U(m)-A(n,D), \tag{4} \]

\[ A(n,D)=\operatorname{Li}(N_2)\,\pi \prod_p\left(1+\frac{\chi_4(p)}{p(p-1)}\right) \prod_{\substack{p>2}} \left(1-\frac{2+\chi_4(p)-\dfrac{1}{p}-\dfrac{\chi_4(p)}{p}}{p^2-p+\chi_4(p)}\right) \times \]
\[ \times \prod_{\substack{p\mid D\\ p>2}} \frac{(p-\chi_4(p))(p-1)}{p^2-p-2+\dfrac{1}{p}+\dfrac{\chi_4(p)}{p}}^{*}. \tag{5} \]

In the present case, the dispersion in progressions is the expression

\[ V=\sum_D (r(n,D))^2 = \sum_D \left( \sum_{\substack{m\le n\\ m=n-D\nu\in(QFR)\\ \nu\ \text{prime}}} U(m)-A(n,D) \right)^2 . \tag{6} \]

Main lemma.

\[ V\ll D_2N_2^2\frac{1}{(\ln n)^C}; \tag{7} \]

here \(C\) is an arbitrarily large constant.

The solution of problem (2) with the aid of the main lemma is quite simple: if \(D\) runs only through prime numbers, then, by discarding the remaining numbers, we only decrease the dispersion, so that

\[ \sum_{\substack{D\ \text{prime}}} \left( \sum_{\substack{m\le n\\ m=n-D\nu\in(QFR)\\ \nu\ \text{prime}}} U(m)-A(n,D) \right)^2 \ll D_2N_2^2\frac{1}{(\ln n)^C}. \tag{8} \]

It follows at once from this that there exist representations
\(n=D\nu+\xi^2+\eta^2\), where \(D\) and \(\nu\) are prime and
\(D_1-D_2\le D<D_1\). Indeed, if there were no such solutions, then the terms
\(\sum U(m)\) in the parentheses in (8) would vanish, and the resulting sum would coincide with

\[ \sum_{\substack{D\ \text{prime}}} (A(n,D))^2>\frac{N_2^2D_2}{\ln^4 n}, \]

which would contradict (8). A simple computation also gives the asymptotic formula (9).

Let us explain the principles of evaluating (6). The parentheses on the left-hand side of (6) are expanded, and the order of summation is changed: first we sum over \(D\). The main difficulty is the summation of the term

\[ \sum_D \left(\sum_\nu U(n-D\nu)\right)^2 = \sum_{\nu_1,\nu_2,D} U(n-D\nu_1)U(n-D\nu_2). \]

Here \(\nu_1,\nu_2\) run through identical sequences of prime numbers;
\(n-D\nu_i\in(QFR)\) \((i=1,2)\).

Further, from elementary considerations we derive, for given \(\nu_1,\nu_2\),

\[ \sum_D U(n-D\nu_1)U(n-D\nu_2)= \]

\[ =\frac14 \sum_{\substack{\delta\in(QFR)\\ \delta\le n}} (U(\delta))^2 \sum_{\substack{t\in(QFR)\\ (t,\delta)=1\\ t\le n}} \mu(t) \sum_{\substack{D\\ n-D\nu_i\in(QFR)\\ t\delta\mid n-D\nu_i\ (i=1,2)}} U\left(\frac{n-D\nu_1}{\delta}\,\frac{n-D\nu_2}{\delta}\right). \tag{9} \]

Thus the matter is reduced to counting, as \(D\) varies, the number of representations

\[ (n-D\nu_1)(n-D\nu_2)=\delta^2(\xi^2+\eta^2) \]

or

\[ n^2(\nu_1-\nu_2)^2 = (4\nu_1\nu_2D-n(\nu_1+\nu_2))^2 - 4\delta^2\nu_1\nu_2(\xi^2+\eta^2), \tag{10} \]

\[ {}^{*}\ \text{For simplicity it is assumed here that } n \text{ has no small divisors.} \]

i.e., the square number \(n^2(\nu_1-\nu_2)^2\) must be represented by the ternary quadratic form \(x^2-4\delta^2\nu_1\nu_2(\xi^2+\eta^2)\) under certain geometric and congruence conditions on \(x\). Such a problem is solved quite satisfactorily (although rather cumbersomely) in the more general theory of the author and A. V. Malyshev (see the literature in note \((^1)\)). The only computational difficulty is the calculation of the corresponding arithmetical factor.

The general problem (1) is treated analogously. The method can also be applied to decompositions of the form

\[ n=N(\mathfrak a)+Q(\xi,\eta), \tag{11} \]

where \(N(\mathfrak a)\) is the norm of an ideal from any (not only Abelian, as in note \((^1)\)) field, and moreover the ideal \(\mathfrak a\) may be taken from any prescribed class. An asymptotic law also holds. This generalizes Kloosterman’s theorem on an equation of the form \(n=ax^2+by^2+cz^2+dt^2\).

Finally, the general solution of equation (1) can be applied to the study of the well-known Hardy–Littlewood equation \((^2)\)

\[ n=p+\xi^2+\eta^2. \tag{12} \]

This, however, requires very laborious computations.

CITED LITERATURE

\(^1\) Yu. V. Linnik, DAN, 120, No. 5, 960 (1958).
\(^2\) H. H. Hardy, J. E. Littlewood, Acta Math., 44 (1923).