Additive function¶

An arithmetic function satisfying f(ab)=f(a)+f(b) whenever positive integers a and b are coprime.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3206
Origin domain: multiplicative number theory
Subdomain: multiplicative number theory

Core Idea¶

Completely additive functions remove the coprimality restriction; additive functions are determined by their values on prime powers and support average-order and probabilistic number theory. Unique prime factorization decomposes a positive integer into coprime prime-power factors, and the addition law converts multiplication of those factors into a sum of assigned values. 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¶

Additive function belongs to multiplicative number theory and is useful where the analyst can specify the typed multiplicative number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the positive-integer domain and codomain, arithmetic-function convention, coprimality condition, value at one, prime-power values, complete-additivity specialization, growth or regularity assumptions, and distinction from Cauchy-additive maps are explicit. The scope is broad within that domain but bounded by the need for the positive-integer domain and codomain, arithmetic-function convention, coprimality condition, value at one, prime-power values, complete-additivity specialization, growth or regularity assumptions, and distinction from Cauchy-additive maps are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the positive-integer domain and codomain, arithmetic-function convention, coprimality condition, value at one, prime-power values, complete-additivity specialization, growth or regularity assumptions, and distinction from Cauchy-additive maps 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. A bare label is insufficient because the name Additive function can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

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 Additive function. Additive function 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 multiplicative number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the positive-integer domain and codomain, arithmetic-function convention, coprimality condition, value at one, prime-power values, complete-additivity specialization, growth or regularity assumptions, and distinction from Cauchy-additive maps are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of multiplicative number theory because they reuse the typed multiplicative number theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Unique prime factorization decomposes a positive integer into coprime prime-power factors, and the addition law converts multiplication of those factors into a sum of assigned values., and type the carrier, state every parameter and convention in the definition, test that the positive-integer domain and codomain, arithmetic-function convention, coprimality condition, value at one, prime-power values, complete-additivity specialization, growth or regularity assumptions, and distinction from Cauchy-additive maps are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Additive function Domain-specific

Parents (1) — more general patterns this builds on

Additive function is a kind of Function (Mapping) Prime

The proposed strict upward parent is prime:function_mapping.

Hierarchy path (1) — routes to 1 parentless root

Additive function → Function (Mapping)

Neighborhood in Abstraction Space¶

Additive function sits in a crowded region of the domain-specific corpus (4^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

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