## A mean value theorem on the binary Goldbach problem and its application.(English)Zbl 1188.11050

Summary: We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers $$n$$ satisfying $$n \not\equiv 2 \pmod 6$$, the equation $$p_{1} + p_{2} = n$$ is solvable in prime variables $$p_{1}, p_{2}$$ such that $$p_{1} + 2 = P_{3}$$, and for every sufficiently large odd integer $$\bar n$$ satisfying $$\bar n \not\equiv 1 \pmod 6$$, the equation $$p_{1} + p_{2} + p_{3} = {\bar n}$$ is solvable in prime variables $$p_{1}, p_{2}, p_{3}$$ such that $$p_{1} + 2 = P_{2}$$, $$p_{2} + 2 = P_{3}$$. Here $$P_{k}$$ denotes any integer with no more than $$k$$ prime factors, counted according to multiplicity.

### MSC:

 11P32 Goldbach-type theorems; other additive questions involving primes 11N36 Applications of sieve methods

### Keywords:

binary Goldbach problem
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### References:

 [8] Tolev DI (1997) On the number of representations of an odd integer as a sum of three primes, one of which belongs to an arithmetic progresson. In: Lupanov OB (ed) Proc Steklov Inst Math 218: 414–432. Moscow: Nauka · Zbl 0911.11048
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