The prime k-tuplets conjecture on average. (English) Zbl 0719.11066

Balog, Antal

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.

Reviewer: D.Wolke (Freiburg i.Br.)

MSC:

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

Keywords:

prime k-tuplets; Bombieri-Vinogradov type theorem; weighted form of the Hardy-Littlewood circle method