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Divisor Function

The multiplicative arithmetic-function family \(\sigma_z(n)=\sum_{d\mid n}d^z\), including divisor count and divisor sum as distinguished cases.

Version
v2 · 2026-09-19 · History
Domain-specific #
1702
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Number Theory, Arithmetic Functions → Mathematics

Core Idea

The divisor functions are the arithmetic-function family

\[ \sigma_z(n)=\sum_{d\mid n}d^z, \]

where \(n\) is a positive integer, the sum ranges over its positive divisors, and \(z\) is real or complex. NIST's Digital Library of Mathematical Functions uses this definition and identifies \(\sigma_0(n)\) with the divisor-counting function and \(\sigma_1(n)\) with the sum of divisors. When the phrase “the divisor function” appears without a parameter, it often means \(d(n)=\tau(n)=\sigma_0(n)\); notation must therefore be declared.

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Ways to Share Fairly

Take 6 cookies. You can share them fairly with 1, 2, 3, or 6 kids, with nothing left over. Counting those ways, 4 of them, is one divisor function, and adding up those numbers, 1 + 2 + 3 + 6 = 12, is another.

Counting and Adding Divisors

A divisor of a number is a number that divides it evenly: the divisors of 6 are 1, 2, 3, and 6. Divisor functions are rules that take a number and do something with all its divisors. One version counts them, so 6 has 4 divisors. Another adds them up: 1 + 2 + 3 + 6 = 12. Other versions first raise each divisor to some power and then add. A neat trick is that you can find these answers by breaking the number into prime numbers first, instead of listing every divisor.

Sigma Divisor-Power Sums

The divisor functions are a family written sigma_z(n): add up d to the power z for every positive divisor d of n. With z = 0 each divisor contributes 1, so sigma_0(n) counts the divisors; with z = 1 you get the sum of the divisors. When someone says the divisor function without a subscript, they often mean the divisor count, sometimes written d(n) or tau(n), so it's important to say which one you mean. The key property is multiplicativity: if m and n share no common factor, then sigma_z(mn) = sigma_z(m) times sigma_z(n). So once you factor n into primes, you can work out each prime-power piece separately and multiply. For example, if n = p^a q^b, the number of divisors is (a+1)(b+1).

 

The divisor functions are the arithmetic functions sigma_z(n) = sum over d dividing n of d^z, for positive integers n and real or complex z. Special cases: sigma_0(n) is the divisor-counting function, also written d(n) or tau(n), and sigma_1(n) is the sum of divisors; since the bare phrase the divisor function often means d(n), notation must be declared. The family aggregates the divisor lattice of an integer into a weighted sum. Its central property is multiplicativity: if gcd(m, n) = 1, then sigma_z(mn) = sigma_z(m) sigma_z(n). Consequently, writing n as a product of prime powers p_i^(a_i) gives sigma_z(n) as the product over i of (1 + p_i^z + p_i^(2z) + ... + p_i^(a_i z)), and tau(n) as the product of (a_i + 1). This reduction from global divisor enumeration to independent prime-power computations is what makes divisor functions part of the foundational machinery of arithmetical functions and Dirichlet multiplication in analytic number theory.

Scope of Application

In elementary number theory, \(\tau(n)\) counts factors, \(\sigma_1(n)\) classifies perfect, deficient, and abundant numbers through comparison with \(2n\), and \(s(n)\) generates aliquot sequences. In multiplicative number theory, the prime-power product supplies exact evaluation and Euler products. In analytic number theory, summatory behavior leads to the Dirichlet divisor problem; DLMF records

\[ \sum_{n\le x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}) \]

Clarity

The parameterized notation resolves recurring ambiguity. “Number of divisors” is \(\sigma_0\), “sum of divisors” is \(\sigma_1\), and “sum of reciprocal divisors” is \(\sigma_{-1}\). Writing \(\sigma_z\) prevents a proof about one specialization from being silently applied to another.

Prime factorization also clarifies mechanism. The value is not mysterious global behavior: every divisor chooses an exponent \(0\le j_i\le a_i\) independently at each prime.

Manages Complexity

Naively, evaluating \(\sigma_z(n)\) suggests scanning integers up to \(n\). Once the factorization \(n=\prod p_i^{a_i}\) is known, the product formula reduces evaluation to \(\sum_i(a_i+1)\)-scale local terms rather than enumerating all candidates. The compression is structural: a divisor corresponds bijectively to one choice of exponent at each prime.

Abstract Reasoning

Multiplicativity: when \(\gcd(m,n)=1\), every divisor of \(mn\) factors uniquely as \(ab\) with \(a\mid m,b\mid n\), so \((ab)^z=a^zb^z\) and the double sum factors. Prime-power evaluation: divisors of \(p^a\) are exactly \(1,p,\ldots,p^a\). Dirichlet convolution: if \(\operatorname{id}_z(n)=n^z\), then \(\sigma_z=\mathbf1*\operatorname{id}_z\), explaining the product of zeta functions in the Dirichlet series.

Knowledge Transfer

Within number theory, the full mechanism transfers literally among exact computation, multiplicative-function proofs, Euler products, average orders, perfect-number questions, and modular forms. The prime-local factorization and divisor aggregation remain unchanged.

Outside this domain, Aggregation and factorwise decomposition are portable parent structures, but the named divisor function is not. In algebraic number theory, analogous ideal-divisor sums are genuine extensions with different objects and norms; in data analysis, “divisor function” has no general transferable meaning.

Relationships to Other Abstractions

Local relationship map for Divisor FunctionParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Divisor FunctionDOMAINPrime abstraction: Aggregation — presupposesAggregationPRIME

Current abstraction Divisor Function Domain-specific

Parents (1) — outgoing typed relations

  • Divisor Function presupposes Aggregation Prime

    Divisor Function presupposes prime:aggregation: each value collapses a complete weighted divisor set to one number.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Divisor Function sits in a sparse region of the domain-specific corpus (62nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

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