Divisor count function

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
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Definition

Let $n$ be a natural number. The divisor count function of $n$ , denoted $d (n)$ or $τ (n)$ , is defined as the number of positive divisors of $n$ . In other words:

$τ (n) = \sum_{d | n} 1$ .

Formula in terms of prime factorization

Suppose we have:

$n = p_{1}^{k_{1}} p_{2}^{k_{2}} \dots p_{r}^{k_{r}}$ .

Then:

$τ (n) = \prod_{i = 1}^{r} (k_{i} + 1)$ .

Behavior

Lower bound

The divisor count function of $n$ takes its lowest value (other than $1$ ) at primes.

$τ (p) = 2 \forall p$ .

In particular:

$lim {inf}_{n \to \infty} τ (n) = 2$ .

Upper bound

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