In mathematics, the Schnirelmann density of a subset A of the set N of non-negative integers is defined as follows: for each integer n > 0, let

Then, the Schnirelmann density of A is

The Schnirelmann density is named after Russian mathematician L. G. Schnirelmann, who was the first to study it; the Schnirelmann density function σ has the following properties:

  1. For all n, A(n) > n . σA.
  2. σA = 1 iff A ⊇ N.
  3. If 1 ∉ A, then σA = 0.
  4. If 0 ∈ A ∩ B, then σ(A + B) ≥ σA + σB - σA . σB
  5. If σA + σB ≥ 1, then σ(A + B) = 1.
  6. If σA > 0, then A is an additive basis.

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