Lemma 2.3. Assuming that:

  • z≥3

  • g:ℕ→[0,1] multiplicative

  • for some K,A∈ℝ we have

    ∑p≤zg(p) log ⁡p≤κ log ⁡z+A.

Then
1∑m≤zh(m)≤2∏p≤z1∕(eκ+1)(1−g(p)),

where h is defined in terms of g as in Selberg’s sieve.