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The prime k-tuplets conjecture on average. (English) Zbl 0719.11066
Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL (USA) 1989, Prog. Math. 85, 47-75 (1990).
[For the entire collection see Zbl 0711.00008.]
H. Maier and C. Pomerance [Trans. Am. Math. Soc. 322, No.1, 201-237 (1990; Zbl 0706.11052)] recently proved a Bombieri-Vinogradov type theorem for prime pairs. Let \[ \Psi (x,n,a,q)=\sum_{m\leq x,\quad m\equiv a(q)}\Lambda (n)\Lambda (m+n), \] then, for some \(\alpha >0\) and any \(A>0\), \[ \sum_{n\leq x,2| n}\sum_{q\leq x^{\alpha}}\max_{a;(a,q)=(a+n,q)=1}| \Psi (x,n,a,q)-\frac{x}{\phi (q)}K\prod_{p| n,p>2}\frac{p-1}{p-2}| \quad \ll_ A\quad \frac{x^ 2}{\ln^ Ax} \] (K\(=2\prod_{p>2}p(p-2)/(p-1)^ 2).\)
In the present paper this is sharpened \((q\leq x^{1/3}(\ln x)^{-B})\) and generalized to k-tuplets. The proof depends on (i) a complicated induction process, beginning with the well known Bombieri-Vinogradov theorem, and (ii) a weighted form of the Hardy-Littlewood circle method, which seems to be of independent interest.
The author announces further results in this context, in particular the following generalizations of the “three primes in progression” theorem. There are infinitely many triangles and squares with prime entries.

11P32 Goldbach-type theorems; other additive questions involving primes
11P55 Applications of the Hardy-Littlewood method
11N05 Distribution of primes