On long κ-tuples with few prime factors

Abstract

We prove that there are infinitely many integers n such that the total number of prime factors of (n + h1) • • • (n + hκ) is exactly (1 + o(1))κ Log κ. Our result even ensures us that these prime factors are fairly evenly distributed among every factors n + hi.

The large sieve revisited

“…It is hoped that this wider perspective will have applications especially to "non-abelian" generalizations of the large sieve. Finally, we point out that Ramaré has developed similar material in his lecture notes [12] (see also [13][14][15]).…”