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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)

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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