Arithmetic function d(n) (also denoted τ(n)) equal to the number of positive divisors of the integer n.

Principle

Multiplicative and computed from the prime-power factorization n = ∏ p_i^{a_i} via d(n) = ∏ (a_i + 1); each exponent a_i contributes a_i + 1 choices for its divisor exponent.

Demonstration

Example: n = 28 = 2^2·7^1, so d(28) = (2+1)·(1+1) = 6; the positive divisors are {1,2,4,7,14,28}.

Misapplication

Counting only distinct prime factors instead of all divisors, or including negative divisors without noting sign conventions, or assuming multiplicativity across non-coprime factors are common mistakes.

Consequence

d(n) appears in divisor-sum identities, Dirichlet convolution relations, and average-order results (it grows slowly on average); it provides a basic measure of an integer's arithmetic complexity and factors into analytic estimates in number theory.

Reversal

Contrast with the sum-of-divisors σ_1(n): while d counts quantity, σ measures total size; alternatively, the indicator of squarefree numbers |μ(n)| distinguishes numbers with maximal 'small' d(n) behaviour from those with higher divisor counts due to repeated prime powers.

Boundary

Defined for positive integers n (extension to other rings requires reinterpretation); multiplicativity presumes prime-power decomposition and is invalid without it. The function counts positive divisors unless an explicit sign convention is given.

Semantic Tension

Often conflated with σ_k functions (especially σ_0 = d) and with functions that count prime divisors; precise distinction is whether one counts divisors (d) or sums them (σ) or counts prime factors (ω, Ω).

Synthesis

The divisor function d(n) = τ(n) is the multiplicative arithmetic function giving the count of positive divisors of n via the formula d(n) = ∏ (a_i+1) from the prime-power factorization; it serves as a simple discrete measure of factor complexity used throughout multiplicative number theory.