Let \(N\) be a sufficiently large even integer, and for an integer \(l_1\), write \(l_2=N-l_1\). When a natural number \(q\) and an integer \(l_1\) satisfy \((l_1l_2,q)=1\), define \(S(N,q)=S(N,q,l_1)\) to be the number of representations of \(N\) in the form \[ N=p+P_2,\qquad p\equiv l_1 \pmod{q},\qquad P_2\equiv l_2 \pmod{q}, \] where \(p\) denotes a prime number, and \(P_2\) denotes an almost prime with at most two prime factors (counted according to multiplicity). Further, let \(\varphi(q)\) denote Euler’s totient function, and put \[ C(N,q)=\prod_{p>2}\bigl(1-(p-1)^{-2}\bigr) \prod_{p|qN, p>2}{p-1 \over p-2}. \] The main theorem of this article states that for all \(q\leq N^{1/37}\) except for \(O(N^{1/37}(\log N)^{-5})\) possible exceptions, one has \[ S(N,q)\geq 0.001C(N,q)N\varphi(q)^{-1}(\log N)^{-2}, \] for all \(l_1\) satisfying \((l_1l_2,q)=1\). The right hand side of the latter inequality coincides with the expected order of magnitude of \(S(N,q)\). The proof actually shows that the number of exceptional values of \(q\) is \(O(N^{1/37}(\log N)^{-A})\) for any fixed \(A\). A similar conclusion is displayed concerning the dual problem, that is, concerning the equation \(p+2=P_2\).
The proof is based on a variant of the famous Chen’s weighted sieve. Estimation of remainder terms is essentially due to the Bombieri-Vinogradov theorem. The exponent \(1/37\) results from evaluation of various constants appearing in the sieve procedure, so that a positive lower bound for \(S(N,q)\) may be obtained at the end. The authors announce that by additional effort, the above conclusion may still be proved with the exponent \(0.028\) in place of \(1/37\).
(The reviewer received a letter from the authors, via the editor, in which they corrected a trifling slip of the pen in this article. Namely, the expression “infinitely many solutions” appears in the abstract and the statements of theorems, but, as is clear from the context, the word “infinitely” should be deleted).