Additive function¶

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

Version: v2 · 2026-09-19 · History
Domain-specific #: 3206
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Multiplicative Number Theory → Mathematics

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.

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Prime Piece Scores

Every whole number can be built by multiplying special building-block numbers called primes, like 12 = 2 × 2 × 3. An additive function gives each group of the same building block a score, and a number's score is the scores of its groups added up. So when you multiply numbers that share no building blocks, their scores just add.

Adding Up Prime Parts

Every whole number bigger than 1 can be made by multiplying prime numbers, like 12 = 2 × 2 × 3, and there's only one way to do it. An additive function is a rule that gives each whole number a value so that when you multiply two numbers with no prime factors in common, the value of the product is the sum of their values. For example, 'how many different primes divide this number' works: 4 has one (2), 9 has one (3), and 36 = 4 × 9 has two. Because of this, you only need to know the function's values on prime powers, like 4, 8, or 9, to know it everywhere. If the adding rule works even when the numbers share primes, it's called completely additive.

Coprime-Additive Arithmetic Function

In number theory, an additive function is an arithmetic function f defined on the positive integers such that f(mn) = f(m) + f(n) whenever m and n are coprime, meaning they share no prime factor. Setting m = n = 1 shows f(1) = 0. Because every positive integer factors uniquely into powers of distinct primes, which are pairwise coprime, f is completely determined by its values on prime powers: f(n) is the sum of f(p^k) over the prime powers in n's factorization. A completely additive function drops the coprimality condition, so f(mn) = f(m) + f(n) for all m and n; for example, the total number of prime factors counted with repetition. These functions are important in studying average behavior and in probabilistic number theory. They should not be confused with Cauchy-additive maps, which satisfy f(x + y) = f(x) + f(y), turning addition, not multiplication, into addition.

An additive function in number theory is an arithmetic function f on the positive integers such that f(mn) = f(m) + f(n) whenever m and n are coprime, meaning they share no prime factor. This forces f(1) = 0. Because every positive integer factors uniquely into powers of distinct primes, which are pairwise coprime, f(n) is the sum of f over those prime powers, so an additive function is completely determined by its values on prime powers. A completely additive function drops the coprimality condition, satisfying f(mn) = f(m) + f(n) for all m and n, so it is determined by its values on primes. This is different from a Cauchy-additive function, which satisfies f(x + y) = f(x) + f(y); an additive arithmetic function turns multiplication, not addition, into addition. Additive functions are central to studying average behavior of arithmetic functions and to probabilistic number theory.

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.
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) — outgoing typed relations

Additive function is a kind of Function (Mapping) Prime

It remains its own entry because its identity is fixed by 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.

Hierarchy path (1) — routes to 1 parentless root

Additive function → is a kind of Function (Mapping)

Neighborhood in Abstraction Space¶

Additive function sits in a crowded region of the domain-specific corpus (2^nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Integer Classifications & Arithmetic Functions (45 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25