A weighted notion of density for a set A of positive integers defined, when it exists, by the limit as x→∞ of (1 / log x) Σ_{n≤x, n∈A} 1/n; it measures the proportion of A with harmonic weighting rather than counting measure.

Principle

Use harmonic weights 1/n to give scale-invariant emphasis to small and large integers so that multiplicative structure and thin sets with divergent reciprocal sums can be detected even when natural density is zero or undefined.

Demonstration

For the set of multiples of an integer m≥1, Σ_{n≤x, n∈A} 1/n ≍ (1/m) log x, so the logarithmic density equals 1/m. This shows how arithmetic progressions receive the expected proportion under harmonic weighting.

Misapplication

Treating logarithmic density as interchangeable with natural (asymptotic) density; assuming existence of the limit for arbitrary sets without checking oscillation of the harmonic sum.

Consequence

When it exists, logarithmic density yields a robust fractional size invariant that is stable under thinning by multiplicative scalings and is useful for sets defined by multiplicative or distributional properties.

Reversal

The complement of A has logarithmic density 1 minus the logarithmic density of A when both densities exist; however, neither side need exist even if the other does.

Boundary

Applies only to subsets of positive integers and only when the harmonic-weighted limit exists; excludes finite sets, and many natural sets have no logarithmic density because the weighted partial sums oscillate.

Semantic Tension

Competes with natural (asymptotic) density: natural density counts integers equally, while logarithmic density weights by 1/n, so sets with many small elements or multiplicative structure can have differing values under the two notions.

Synthesis

Logarithmic density is the harmonic-weighted analogue of natural density: it replaces equal-counting by 1/n weights to detect multiplicative and scale-sensitive distributional features of integer sets when the corresponding harmonic average converges.