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