On Bombieri's asymptotic sieve

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 ${\sum }_{n\le x}{a}_n{\mathrm{\Lambda }}_k(n)$ 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.