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UDC 511
MATHEMATICS
T. M. FEDULOVA
APPLICATION OF THE DISPERSION METHOD IN ADDITIVE PROBLEMS WITH A RESTRICTED SET OF PRIME NUMBERS
(Presented by Academician Yu. V. Linnik on 23 VII 1969)
1. The dispersion method, developed by Yu. V. Linnik (¹), is applicable to the solution of various binary additive problems of the form
\[ \alpha+\beta=n, \tag{1} \]
\[ \alpha-\beta=a, \tag{2} \]
where \(\alpha\) and \(\beta\) run through prescribed sequences of natural numbers, sufficiently well distributed in arithmetic progressions; \(a\ne 0\) is a fixed integer; \(n\) is a sufficiently large natural number \((n\to\infty)\), \(\beta<n\) (see (³)).
In particular, the dispersion method makes it possible to find an asymptotic formula for the number of solutions of equations (1) and (2) in the case when \(\alpha\) runs through the sequence of natural numbers with a restricted number of prime divisors, while \(\beta\) takes the values of a given quadratic form.
Equations (1) and (2) can be written in a uniform way in the form
\[ \alpha+(-1)^{j-1}\beta=n_j\qquad (j=1,2), \tag{3} \]
where
\[ n_j= \begin{cases} n, & \text{if } j=1,\\ a, & \text{if } j=2. \end{cases} \]
The aim of this note is to investigate equation (3) for \(\alpha=p_1p_2\ldots p_k\), where \(p_i\) are prime numbers and \(k\) is a fixed natural number. In the case when \(\beta=xy\) and \(\beta=x^2+y^2\), equation (3) is solved by the dispersion method. If \(\beta=\varphi(x,y)\), where \(\varphi(x,y)\) is a positive definite quadratic form with discriminant different from a perfect square, then the asymptotic formula for the number of solutions of equation (3) is found by a combined application of the dispersion method and elementary ergodic considerations (see (², ⁴)).
The results of the note are a generalization of investigations by Yu. V. Linnik (¹), B. M. Bredikhin (³, ⁵, ⁶), and A. A. Polyanskii (⁷) on the problems of Titchmarsh and Hardy–Littlewood.
In what follows we shall assume that \(\alpha=p_1p_2\ldots p_k\) consists of distinct prime numbers, and moreover \((\alpha,n_j)=1\). The general case reduces to this one.
2. Let us formulate the main results. Let \(Q_j(n)\) be the number of solutions of the equation
\[ p_1p_2\ldots p_k+(-1)^{j-1}xy=n_j\qquad (j=1,2), \tag{4} \]
where \(p_1,p_2,\ldots,p_k\) independently run through distinct prime numbers subject to the condition \((p_1p_2\ldots p_k,n_j)=1\), \(k\) is a fixed number, and \(x\) and \(y\) independently run through natural numbers, \(xy<n\).
Theorem 1. As \(n\to\infty\),
\[ Q_j(n)= \frac{315\zeta(3)}{2\pi^4} \prod_{p\mid n_j} \frac{(p-1)^2}{p^2-p+1}\, A_k(n,n_j)\,n + O\left(\frac{n}{(\ln n)^{1-\varepsilon_1}}\right), \tag{5} \]
where
\[ A_k(n,n_j)= \sum_{l=0}^{k-1} c_k^l \sum_{\substack{\delta_1/n_j\\ 0\leqslant \omega(\delta_1)\leqslant l}} \frac{\mu(\delta_1)}{\delta_1} \sum_{\substack{(\delta_2,n_j)=1\\ \omega(\delta_2)=l-\omega(\delta_1)}} \frac{\mu(\delta_2)}{\varphi^2(\delta_2)} \sum_{r=0}^{k-l-1} \frac{B_r}{(k-l-r-1)!}\times (\ln\ln n)^{k-l-r-1}, \tag{6} \]
\(B_r\) are certain constants. The numbers \(B_r\) appear in the asymptotic evaluation of sums of the form:
\[ \sum_{p_1p_2\ldots p_s<n} 1 \quad \text{(see (8));} \]
\(\omega(\delta)\) is the number of distinct prime divisors of \(\delta\).
Let \(S_j(n)\) be the number of solutions of the equation
\[ p_1p_2\ldots p_k+(-1)^{j-1}(x^2+y^2)=n_j,\qquad (j=1,2), \tag{7} \]
where all the conditions from (4) are retained and \(x^2+y^2<n\).
Theorem 2. As \(n\to\infty\),
\[ S_j(n)= \pi \prod_p\left(1+\frac{\chi_4(p)}{p(p-1)}\right) \prod_{p/n_j} \frac{(p-1)(p-\chi_4(p))}{p^2-p+\chi_4(p)} \,B_k(n,n_j)\,\frac{n}{\ln n} + O\left(\frac{n}{(\ln n)^{1+\gamma}}\right), \tag{8} \]
where
\[ B_k(n,n_j)= \sum_{l=0}^{k-1} c_k^l \sum_{\substack{\delta_1/2n_j\\ 0\leqslant \omega(\delta_1)\leqslant l}} \frac{\mu(\delta_1)}{\delta_1} \sum_{\substack{(\delta_2,2n_j)=1\\ \omega(\delta_2)=l-\omega(\delta_1)}} \frac{\mu(\delta_2)}{\delta_2} \sum_{r=0}^{k-l-1} \frac{B_r}{(k-l-r-1)!}\times (\ln\ln n)^{k-l-r-1},\qquad \gamma=0.042. \tag{9} \]
Let us note that the numbers \(A_k(n,n_j)\) and \(B_k(n,n_j)\) are equal to 1 for \(k=1\). Thus, the theorems of Yu. V. Linnik (1), which provide a solution of the Titchmarsh and Hardy–Littlewood problems, and the analogue of the Titchmarsh problem considered in the work of B. M. Bredikhin (3), are obtained as special cases of Theorems 1 and 2 for \(k=1\).
Let \(Q(n)\) be the number of solutions of the equation
\[ p_1p_2\ldots p_k+\varphi(x,y)=n, \tag{10} \]
where \(\varphi(x,y)\) is a positive definite quadratic form with discriminant different from a perfect square, and the conditions imposed on the numbers \(p_i\) in (4) are retained.
Theorem 3. As \(n\to\infty\),
\[ Q(n)=\sigma_k(n,\varphi)\frac{n}{\ln n} + O\left(\frac{n}{(\ln n)^{1+\gamma_1}}\right), \tag{11} \]
where \(\sigma_k(n,\varphi)\gg 1/(\ln\ln n)^C\) is a complicated arithmetic factor, \(0<\gamma_1<\gamma\).
3. Let us consider brief outlines of the proofs of Theorems 1–3.
1) We transform \(Q_j(n)\), using the symmetry of equation (4) with respect to \(x\) and \(y\), as well as upper estimates for \(\pi_k(n)\) (8). It is easy to see that
\[ Q_j(n)=2\widetilde Q_j(n)+O\bigl(n(\ln n)^{-1+\varepsilon_2}\bigr), \tag{12} \]
where \(\widetilde Q_j(n)\) is the number of solutions of equation (4) under the additional condition
\(x<\sqrt n\,n_1^{-1}\), \(n_1=\exp(\ln n)^{\varepsilon_0}\), \(\varepsilon_0>0\) a small constant. In this case
\[ \widetilde Q_j(n)=\sum_{s=1}^k C_k^s Q_s(n), \tag{13} \]
where \(Q_s(n)\) is the number of solutions of equation (4) with \(x<\sqrt n\,n_1^{-1}\), when among the numbers \(p_1,p_2,\ldots,p_k\) there are \(s\) numbers \(q_1,q_2,\ldots,q_s\), each of which is greater than \(\exp(\ln n)^{\alpha_0}\), \(\alpha_0>0\) a sufficiently small constant, while the remaining \(p_i\leq\)
\[
\leqslant \exp(\ln n)^{\alpha_0}.
\]
Thus, \(Q_s(n)\) is the number of solutions of the equation
\[ p_{i_1}p_{i_2}\ldots p_{i_{k-s}}q_1q_2\ldots q_s+(-1)^{j-1}xy=n_j \qquad (j=1,2) \tag{14} \]
under the conditions indicated above.
In order to reduce equation (14) to a form convenient for applying the dispersion method, we divide the numbers \(q_1q_2\ldots q_s\) into two classes:
I. \(q_1q_2\ldots q_s\) has at least one divisor \(q_i\) such that
\[ \exp(\ln n)^{\alpha_0}<q_i\leqslant P_0(n), \tag{15} \]
where
\[
P_0(n)=\exp\ln n\frac{\ln\ln\ln n}{K_0\ln\ln n},
\]
\(K_0\) is a sufficiently large constant.
II. \(q_1q_2\ldots q_s\) has no divisors \(q_i\leqslant P_0(n)\), i.e. all \(q_i>P_0(n)\).
Let \(q_1q_2\ldots q_s\in \mathrm{I}\). Taking for \(v\) the least of the \(q_i\) satisfying (15), we write equation (14) in the form
\[ vD' + (-1)^{j-1}xy=n_j \qquad (j=1,2), \tag{16} \]
where \(D'\in(D)\), \(v\in(v)\). Equation (16) is solved by the ordinary dispersion method according to the scheme of work \((^3)\).
If \(q_1q_2\ldots q_s\in \mathrm{II}\), then equation (14), with the aid of a generalization of the elementary sieve (see \((^1),(0,6,16)\)), is reduced to equations of the form
\[ p_{i_1}\ldots p_{i_{k-s}}x'_1\ldots x'_{k_1}\ldots z'_1\ldots z'_{k_s} +(-1)^{j-1}xy=n_j, \tag{17} \]
where \(x'_1,\ldots,x'_{k_1},\ldots,z'_1,\ldots,z'_{k_s}\) have no prime divisors \(\leqslant P_0(n)\). Equation (17) is solved with the aid of the ordinary dispersion method when \(k_1+\ldots+k_s\geqslant 7\), and of the complicated dispersion method when \(k_1+\ldots+k_s\leqslant 7\) (see \((^3)\)).
2) We represent \(S_j(n)\), following C. Hooley \((^9)\) and B. M. Bredikhin \((^3)\), as follows:
\[ \begin{aligned} S_j(n)=&\;4 \sum_{p_1\ldots p_k+(-1)^{j-1}2^\lambda xy=n_j}\chi_4(x) -8 \sum_{\substack{p_1\ldots p_k+(-1)^{j-1}2^\lambda xy=n_j\\ x\leqslant \sqrt n\, n_1^{-1}}}\chi_4(x) \\ &-4 \sum_{\substack{p_1\ldots p_k+(-1)^{j-1}2^\lambda xy=n_j\\ \sqrt n\, n_1^{-1}<x<\sqrt{nn_1}\\ y<\sqrt{nn_1}}}\chi_4(x) +4 \sum_{\substack{p_1\ldots p_k+(-1)^{j-1}2^\lambda xy=n_j\\ \sqrt n\, n_1^{-1}<x<\sqrt{nn_1}}}\chi_4(x) \\ &+O\!\left(\frac{n}{(\ln n)^C}\right) =\Sigma_A-\Sigma_{B_1}+\Sigma_{B_2} +O\!\left(\frac{n}{(\ln n)^C}\right), \end{aligned} \tag{18} \]
where \(C\) is a large constant. \(\Sigma_A\) is calculated by the dispersion method according to the scheme of the proof of Theorem 1 (with certain complications), while \(\Sigma_{B_1}\) and \(\Sigma_{B_2}\) are estimated by C. Hooley’s method \((^9)\) according to the scheme developed in \((^3)\).
3) Equation (10) is first solved for the case when the values of \(\varphi(x,y)\) run through a whole sequence of quadratic forms. In doing so, the dispersion method is applied according to the scheme of work \((^3)\), taking into account the considerations expressed in solving equation (4). The transition to an individual form is carried out according to the scheme of work \((^4)\).
Kuibyshev Pedagogical Institute
named after V. V. Kuibyshev
Received
18 VII 1969
CITED LITERATURE
- Yu. V. Linnik, The Dispersion Method in Binary Additive Problems, L., 1961.
- Yu. V. Linnik, Ergodic Properties of Algebraic Fields, L., 1967.
- B. M. Bredikhin, UMN, 20, issue 3 (122), 89 (1965).
- B. M. Bredikhin, Yu. V. Linnik, Mathematics Collection, 71 (113): 2, 145 (1966).
- B. M. Bredikhin, Izv. AN SSSR, Ser. Mat., 27, 2, 439 (1963).
- B. M. Bredikhin, Izv. AN SSSR, Ser. Mat., 27, 3, 577 (1963).
- A. A. Polyanskii, DAN, 180, No. 1, 29 (1968).
- L. G. Sathe, J. Indian Math. Soc., 17, No. 2, 63 (1953).
- C. Hooley, Acta Math., 97, 189 (1957).