Schnirelmann density

Definition

Let $S$ be a subset of the set $\mathbb{N}$ of natural numbers. The Schnirelmann density of $S$ is defined as:

${inf}_{n\in \mathbb{N}}\frac{S\cap \{1,2,\dots ,n\}}{n}$.

The definition can be modified to consider subsets of ${\mathbb{N}}_0$, the natural numbers with zero. the Schnirelmann desity of ${\mathbb{N}}_0$ is defined as:

${inf}_{n\in {\mathbb{N}}_0}\frac{S\cap \{0,1,\dots ,n\}}{n+1}$.

Note that Schnirelmann density is highly sensitive to initial values, and for this reason, the value of the Schnirelmann density depends on whether we are treating the set as a subset of $\mathbb{N}$ or of ${\mathbb{N}}_0$.

Related notions

• Upper asymptotic density
• Lower asymptotic density

Facts

• Mann's theorem
• Schnirelmann's theorem on Goldbach's conjecture