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The ternary Goldbach problem in arithmetic progressions. (English) Zbl 0889.11035
The main result of the paper (see Theorem 1) is the following. Let $$N$$ be a large odd integer and $$R=N^{1/8- \varepsilon}$$. Then for all integers $$r\leq R$$ but $$O(R \log^{-A} R)$$ exceptions, $$N$$ can be written as $N=p_1 +p_2 +p_3$ with primes $$p_j \equiv b_j \pmod r$$, $$j=1,2,3$$. Here $$\varepsilon>0$$ is arbitrarily small, $$A>0$$ is arbitrarily large and the $$b_j$$ satisfy some natural arithmetic conditions.
The interest of this result comes from the uniformity in $$r$$, which is achieved by means of a major-arcs mean-value estimate for the exponential sum over primes, see Theorem 3. Earlier versions of such mean-value estimate were given by D. Wolke [Acta Math. Hung. 61, No. 3-4, 241-258 (1993; Zbl 0790.11075)]. Proofs are based on a combination of the circle method, Vaughan’s identity and analytic methods.
Reviewer: A.Perelli (Genova)

##### MSC:
 11P32 Goldbach-type theorems; other additive questions involving primes 11L07 Estimates on exponential sums 11P55 Applications of the Hardy-Littlewood method
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