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 ∑ _n≤xa _nΛ _k(n) for k ≥ 2. By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.