Dedekind psi function¶

The multiplicative arithmetic function ψ(n)=n times the product of (1+1/p) over the distinct prime divisors of n.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4067
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Number Theory → Mathematics

Core Idea¶

The Dedekind psi function assigns ψ(n)=n∏p|n(1+1/p), equivalently multiplying each prime power p^a by p^a+p(a-1). Prime factorization exposes independent prime-power contributions, making the function multiplicative on coprime inputs. 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.

The load-bearing residual is not the broad topic of number theory. It is It differs from Euler's totient by the plus rather than minus factor and from divisor sums except on prime powers in specific forms..

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The Building-Block Grow Machine

Every counting number is built by multiplying special building-block numbers called primes, like 6 is 2 times 3. The Dedekind psi function is a number machine that looks at which different building blocks a number uses and makes it a bit bigger for each one. Put in 5 and you get 6. Put in 6 and you get 12.

The Plus-One-Over-Prime Rule

Every whole number can be broken into prime numbers multiplied together, like 12 = 2 × 2 × 3. The Dedekind psi function takes a number n and multiplies it by (1 + 1/p) once for each different prime p that divides it. So for 6, which uses primes 2 and 3, you get 6 × 3/2 × 4/3 = 12. For a prime number p, the answer is just p + 1. It's like a cousin of Euler's totient function, which uses (1 − 1/p) instead, so the psi function uses plus where the totient uses minus.

The Totient's Plus-Sign Twin

The Dedekind psi function is defined by ψ(n) = n times the product, over the distinct primes p dividing n, of (1 + 1/p), with ψ(1) = 1. Equivalently, factor n into prime powers and replace each prime power p^a by p^a + p^(a−1), then multiply the results. Because each prime contributes independently, ψ is multiplicative: ψ(mn) = ψ(m)ψ(n) whenever m and n share no prime factor. It looks like Euler's totient function φ(n), which uses (1 − 1/p) instead of (1 + 1/p); psi uses a plus where the totient uses a minus. It also should not be confused with the sum-of-divisors function, which agrees with it only in special prime-power cases. The product is taken over distinct primes, not counted with repetition.

The Dedekind psi function is the arithmetic function ψ(n) = n ∏_{p|n} (1 + 1/p), with the product over distinct prime divisors of n and ψ(1) = 1. Equivalently, on a prime power, ψ(pᵃ) = pᵃ + pᵃ⁻¹, and ψ is determined on all n by multiplying these prime-power values. Prime factorization shows the contributions of distinct primes are independent, so ψ is multiplicative on coprime arguments. Its distinguishing feature relative to Euler's totient φ(n) = n∏(1 − 1/p) is the plus rather than minus factor. It also differs from the divisor-sum function σ except on prime powers in specific forms, such as primes p, where ψ(p) = σ(p) = p + 1. Recognizing an instance means checking the product runs over distinct prime divisors and respects ψ(1) = 1 and coprime multiplicativity.

Scope of Application¶

Dedekind psi function belongs to number theory and is useful where the analyst can specify a positive integer, its distinct prime divisors, the empty-product convention, prime-power factors, and the resulting positive integer value, then evaluate the product ranges over distinct prime divisors and respects ψ(1)=1 and coprime multiplicativity. The scope is broad within that domain but bounded by the need for the product ranges over distinct prime divisors and respects ψ(1)=1 and coprime multiplicativity. 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 product ranges over distinct prime divisors and respects ψ(1)=1 and coprime multiplicativity 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 Dedekind psi 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 Dedekind psi function. Dedekind psi 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: a positive integer, its distinct prime divisors, the empty-product convention, prime-power factors, and the resulting positive integer value. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the product ranges over distinct prime divisors and respects ψ(1)=1 and coprime multiplicativity independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of number theory because they reuse a positive integer, its distinct prime divisors, the empty-product convention, prime-power factors, and the resulting positive integer value, Prime factorization exposes independent prime-power contributions, making the function multiplicative on coprime inputs., and type the carrier, state every parameter and convention in the definition, test that the product ranges over distinct prime divisors and respects ψ(1)=1 and coprime multiplicativity, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Dedekind psi function Domain-specific

Parents (1) — outgoing typed relations

Dedekind psi function is a kind of Function (Mapping) Prime

What makes it its own entry: It differs from Euler's totient by the plus rather than minus factor and from divisor sums except on prime powers in specific forms.

Hierarchy path (1) — routes to 1 parentless root

Dedekind psi function → is a kind of Function (Mapping)

Neighborhood in Abstraction Space¶

Dedekind psi function sits in a crowded region of the domain-specific corpus (11^th 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