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The prime \(k\)-tuplets in arithmetic progressions. (English) Zbl 0797.11076

Let \(k \geq 2\), \(a_ j\) be non-zero integers and \(b_ j\) be integers for \(0 \leq j \leq k-1\). Put \({\mathfrak a} = (a_ 0, \dots, a_{k-1}, b_ 0)\), \({\mathfrak b} = (b_ 1, \dots, b_{k-1})\), \[ N(x,{\mathfrak b}) = \{n:1 \leq a_ j n + b_ j \leq x, \quad j = 0,\dots, k-1\}, \]
\[ \Psi (x,{\mathfrak b},a,q) = \sum_{{n \in N (x,{\mathfrak b}) \atop n \equiv a \pmod q}} \prod^{k-1}_{j=0} \wedge (a_ jn + b_ j), \] \(Z(x) = \{b:| N (x,{\mathfrak b}) | \neq 0\}\) and consider the inequality \[ \sum_{q \leq Q} \max_{1 \leq a \leq q} \sum_{{\mathfrak b} \in Z(x)} | \Psi (x,{\mathfrak b}, a,q) - \text{ expected main term} | \ll x^ k (\log x)^{-A} \tag{1} \] for fixed \({\mathfrak a}\) and any fixed \(A>0\). Using the circle method, A. Balog [Analytic number theory, Prog. Math. 85, 47-75 (1990; Zbl 0719.11066)] proved that (1) holds for any \(h \geq 2\) when \(Q \leq x^{1/3} (\log x)^{-B}\), \(B=B(A)>0\), and H. Mikawa [Tsukuba J. Math. 10, 377-387 (1992; Zbl 0778.11053)] improved Balog’s result to \(Q \leq x^{1/2} (\log x)^{-B}\), \(B = B(A)>0\), in the case \(h=2\).
The author extends Mikawa’s result to the general case \(k \geq 2\), i.e. he proves (1) for any fixed \({\mathfrak a}\) and \(A>0\), \(k \geq 2\) and \(Q \leq x^{1/2} (\log x)^{-B}\), \(B = B(A,k)>0\). He also proves a short intervals version of (1), where \(N(x,{\mathfrak b})\) is replaced by \(N(x,y,{\mathfrak b}) = \{n:x-y<a_ jn + b_ j\leq x\), \(j = 0,\dots, k-1\}\) and analogously for \(\Psi (x,y, {\mathfrak b}, a,q)\) and \(2(x,y)\), and the right hand side of (1) is replaced by \(y^ k (\log x)^{-A}\), provided \(x^{2/3} (\log x)^ c<y \leq x\) and \(Q \leq yx^{-1/2} (\log x)^{- B}\), \(B=B(A,k)>0\).
Reviewer: A.Perelli (Genova)

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11N13 Primes in congruence classes
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