Representation of even numbers by the sum of two odd prime numbers in an arithmetic progression (Q1603075)

Let \(D=p ^{\nu}\), where \(p>2\) is prime and \(\nu\) is a positive integer. Let \(R(n)\) denote the number of representations of \(n\) as a sum of two primes from two arithmetic progressions: \[ n=p_1+p_2, \quad p_i \equiv l_i \bmod D, \quad (l_i,D)=1, \;i=1,2. \] Let \(E_D(X)\) denote the number of integers \(n \leq X\) which cannot be represented in this form. The author proves that for \(D \ll \ln^A x\) and with the exception of \(E_D(X) \ll \frac{X}{\varphi(D) \exp( c_1 ( \log X)^{1/2})}\) integers \(n \leq X\) one has \[ R(n) \gg \frac{n}{\varphi(D) \ln^2 n} (1- \frac{\ln^A n}{\exp( c_2 ( \log X)^{1/2})}) \frac{1}{\exp(\frac{c_2}{4} ( \log X)^{1/2})}, \] where \(c_1\) and \(c_2\) are constants that do not depend on \(A\). This improves a result by \textit{A. F. Lavrik} [Vestn. Leningr. Univ. 16, Mat. Mekh. Astron. No. 3, 11-27 (1961; Zbl 0116.03701)], which proved an asymptotic formula for \(R(n)\) but with \(E_D(X) \ll x \ln^{-A} X\) exceptions. In the special case \(D=1\) it is known that the number of exceptions is at most \(E(X) \ll X^{1- \delta}\) [\textit{I. Allakov}, Vopr. Vychisl. Prikl. Mat. 77, 37-41 (1985; Zbl 0581.10022), \textit{H. L. Montgomery} and \textit{R. C. Vaughan}, Acta Arith. 27, 353-370 (1975; Zbl 0301.10043)].