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.
Core Idea¶
Jordan's totient function J_k(n) counts ordered k-tuples whose coordinates together with n have gcd one. It equals n^k times ∏(1−p^(−k)) over primes dividing n and reduces to Euler's totient at k=1. For each prime p dividing n, a fraction 1/p^k of k-tuples is excluded because every coordinate is divisible by p. For each prime p dividing n, a fraction 1/p^k of k-tuples is excluded because every coordinate is divisible by p.
Scope of Application¶
The function appears in multiplicative number theory, primitive residue counts, Dirichlet convolution, and orders of matrix groups over residue rings. Use it in primitive residue counts, multiplicative identities, Möbius inversion, Dirichlet series, and matrix-group orders with positive k, modulus n, tuple order, and joint-coprimality convention explicit.
- 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. The closest near miss sets the boundary: Euler's totient is the closest near miss and exact k=1 member; using it for higher-dimensional primitive tuples omits local degrees of freedom.
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. The central combinatorial count–closed product tradeoff is this: Enumeration displays the objects while factorization makes large cases tractable. A second joint primitiveness–coordinate intuition tension matters because A tuple may be primitive although no coordinate is individually coprime to n.
Abstract Reasoning¶
Use three linked moves: 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. As a collapse test, the case exits when k is not a positive integer, n is outside the stated domain, or the gcd condition is changed. A fourth check is to use the product or divisor-convolution identity for computation. A final check is to 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. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. J_k measures a finite class of primitive tuples. Prime-local conditions assemble multiplicatively.
Relationships to Other Abstractions¶
Current abstraction Jordan's totient function Domain-specific
Parents (1) — more general patterns this builds on
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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 (57th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Fermat's Little Theorem — 0.88
- Square-Free Integer — 0.86
- Covering Set — 0.85
- Automorphic number — 0.85
- Regular prime — 0.84
Computed from structural-signature embeddings · 2026-10-08