Jordan's totient function¶

The multiplicative arithmetic function J_k(n) counting ordered k-tuples modulo n that are jointly coprime with n, equivalently n^k times the product of 1−p^(−k) over primes dividing n.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10193
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory, Arithmetic Functions → Mathematics

Core Idea¶

Jordan's totient function J_k(n) counts ordered k-tuples of residues whose coordinates are not all divisible by any prime dividing n—equivalently, whose coordinates together with n have greatest common divisor one. The k=1 case is Euler's totient.

For each prime p dividing n, a fraction 1/p^k of k-tuples is excluded because every coordinate is divisible by p. Multiplying the surviving factors gives J_k(n)=n^{k∏_{p|n}(1−p}(−k)), making multiplicativity transparent.

Divisor sums, Möbius inversion, Dirichlet series, and matrix-group order formulas expose the function's reach. Those results depend on the precise joint-coprimality count; ratios such as Dedekind's psi are related arithmetic functions, not alternate names.

Structural Signature¶

Sig role-phrases:

modulus n. Sets the finite residue universe and prime divisors. Constitutive parameter. If altered: Nonpositive or unstated modulus leaves the count undefined.
dimension k. Fixes tuple length and the exponent in each local factor. Constitutive family index. If altered: Changing k changes the function rather than one evaluation.
primitive tuple condition. Requires no prime divisor of n to divide every coordinate. Identity-bearing predicate. If altered: Coordinatewise coprimality is stronger and counts something else.
prime-local exclusion. Removes the p^k residue tuples simultaneously divisible by each p|n. Constitutive product mechanism. If altered: Ignoring one prime overcounts nonprimitive tuples.
multiplicative assembly. Combines coprime moduli and supports divisor, convolution, and group-order identities. Characteristic arithmetic structure. If altered: Additivity would contradict the prime-factor construction.

What It Is Not¶

Not coordinatewise coprimality. Only the coordinates collectively must avoid a common prime divisor with n.
Not Euler phi except k=1. Higher dimensions count tuples.
Not Dedekind psi. Psi is a ratio involving J_2 and J_1.
Not an arbitrary multiplicative function. The primitive-tuple interpretation fixes local factors.

Scope of Application¶

The function appears in multiplicative number theory, primitive residue counts, Dirichlet convolution, and orders of matrix groups over residue rings.

Tuple counting. Counts primitive k-vectors modulo n.
Multiplicative functions. Factors over coprime moduli.
Möbius inversion. Recovers J_k from the divisor sum n^k.
Dirichlet series. Relates zeta(s−k) to zeta(s).
Finite matrix groups. Enters GL, SL, and symplectic order formulas.

Clarity¶

The phrase ‘coprime tuple’ should be expanded to gcd(a1,…,ak,n)=1. This permits an individual coordinate to share a factor with n so long as no prime divides all coordinates and n.

Manages Complexity¶

J_k compresses an n^k search space into a product over distinct prime divisors. The same local exclusion structure explains multiplicativity, convolution identities, and matrix-group counts.

Abstract Reasoning¶

Fix positive integers k and n and factor n into distinct primes.
Interpret primitiveness as the absence of one prime dividing every coordinate.
Exclude the p-divisible sublattice for each p and combine by local independence.
Use the product or divisor-convolution identity for computation.
Check whether a proposed application needs ordered tuples and joint, rather than coordinatewise, coprimality.

Knowledge Transfer¶

The local-to-global exclusion pattern transfers to primitive lattice points and finite-module counts. The name J_k remains specific to this arithmetic function and should not label every higher-dimensional totient analogue.

Examples¶

Canonical¶

For n=p prime, exactly the all-zero residue vector fails primitiveness, so J_k(p)=p^k−1, matching p^k(1−p(−k)).

Mapped back: modulus n → prime p; dimension k → tuple length k; primitive tuple condition → not all zero mod p; prime-local exclusion → one p-divisible vector; multiplicative assembly → single local factor.

Applied / In Practice¶

The documented formula for |GL(m,Z/n)| multiplies powers of n by J_1(n) through J_m(n), using successive choices of primitive independent columns over the residue ring.

Mapped back: modulus n → residue ring Z/n; dimension k → successive column stage; primitive tuple condition → admissible primitive column; prime-local exclusion → noninvertible local choices removed; multiplicative assembly → product of Jordan factors.

Structural Tensions¶

T1: combinatorial count vs. closed product. Enumeration displays the objects while factorization makes large cases tractable. Diagnostic: Which representation best supports the proof or computation?

T2: joint primitiveness vs. coordinate intuition. A tuple may be primitive although no coordinate is individually coprime to n. Diagnostic: Is gcd taken collectively?

T3: family generality vs. special-function identity. J_k unifies dimensions while k-specific ratios and group formulas have additional structure. Diagnostic: Which k and application theorem are actually in force?

Structural–Framed Character¶

Jordan's totient is structural. Counting, prime localization, and multiplicativity are formal; conventions are minimal. Its portable skeleton is Counting, but no strict parent edge is added because the current repair preserves frozen placement. Evaluative and practice dependence are negligible; institutional origin is mathematical; vocabulary travels through exact isomorphism only; metaphorical use is inappropriate. Its character: a local-to-global count of primitive residue tuples.

Structural Core vs. Domain Accent¶

Skeletal core. Exclude locally forbidden tuples and multiply the surviving proportions across independent prime components.

Domain-bound accent. Positive integers, gcd, residues, primes, convolution, and finite groups define J_k.

Why not prime. Local exclusion counting travels, but this named function is number-theoretic.

This entry is a kind of Function (Mapping).

Counting. J_k measures a finite class of primitive tuples.
Factorization. Prime-local conditions assemble multiplicatively.
No strict DAG edge is asserted.

Relationships to Other Abstractions¶

Current abstraction Jordan's totient function Domain-specific

Parents (1) — more general patterns this builds on

Jordan's totient function is a kind of Function (Mapping) Prime

It is an arithmetic function.

Hierarchy path (1) — routes to 1 parentless root

Jordan's totient function → Function (Mapping)

Neighborhood in Abstraction Space¶

Jordan's totient function sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Euler totient. Tell: Is k=1 or a k-tuple being counted?
Dedekind psi. Tell: Is the function J_k or a ratio of Jordan totients?
Primitive vector. Tell: Which modulus and joint gcd condition define primitiveness?
Möbius function. Tell: Is inversion machinery or the counted function itself meant?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Jordan%27s_totient_function (revision 1342257477).
Preserved source candidate: https://eudml.org/doc/126410
Preserved source candidate: http://www.math.hmc.edu/~orrison/research/papers/totient.pdf
Preserved source candidate: https://web.archive.org/web/20160305132124/https://www.math.hmc.edu/~orrison/research/papers/totient.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.