Dirichlet density¶

An analytic density of a set of primes defined by its weighted prime Dirichlet series as the exponent approaches one from above.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4197
Origin domain: analytic number theory
Subdomain: analytic number theory
Aliases: Analytic density

Core Idea¶

Existence of natural density implies the same Dirichlet density under standard conditions, but the converse can fail; upper and lower densities handle nonexistent limits. Primes are weighted by inverse powers p to the s, the subset sum is normalized by the corresponding all-prime divergence and the limiting share near s equals one defines density. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Dirichlet density belongs to analytic number theory and is useful where the analyst can specify the typed analytic number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the subset of primes, real parameter s greater than one, weighted subset and all-prime sums, normalization, one-sided limit, upper and lower variants and comparison with natural density or Euler-product singularity are explicit. The scope is broad within that domain but bounded by the need for the subset of primes, real parameter s greater than one, weighted subset and all-prime sums, normalization, one-sided limit, upper and lower variants and comparison with natural density or Euler-product singularity are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the subset of primes, real parameter s greater than one, weighted subset and all-prime sums, normalization, one-sided limit, upper and lower variants and comparison with natural density or Euler-product singularity are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Dirichlet density. Dirichlet density compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed analytic number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the subset of primes, real parameter s greater than one, weighted subset and all-prime sums, normalization, one-sided limit, upper and lower variants and comparison with natural density or Euler-product singularity are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of analytic number theory because they reuse the typed analytic number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Primes are weighted by inverse powers p to the s, the subset sum is normalized by the corresponding all-prime divergence and the limiting share near s equals one defines density., and type the carrier, state every parameter and convention in the definition, test that the subset of primes, real parameter s greater than one, weighted subset and all-prime sums, normalization, one-sided limit, upper and lower variants and comparison with natural density or Euler-product singularity are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Dirichlet density Domain-specific

Parents (1) — more general patterns this builds on

Dirichlet density is a kind of Measure Prime

The proposed strict upward parent is prime:measure.

Hierarchy paths (2) — routes to 2 parentless roots

Dirichlet density → Measure → Aggregation → Micro Macro Linkage

Show alternative path (1)

Neighborhood in Abstraction Space¶

Dirichlet density sits in a crowded region of the domain-specific corpus (16^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Arithmetic Functions & Number Sequences (16 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08