On Bombieri's asymptotic sieve

Abstract

If a sequence $(a_n)$ of non-negative real numbers has ``best possible'' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum ${\sum }_{n\le x}{a}_n{\Lambda }_k(n)$ for $k\ge 2$. By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.

13 pages. To appear in Trans. Amer. Math. Soc. http://www.math.uiuc.edu/~ford/