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 $\tau (n)$, is defined as the number of positive divisors of $n$. In other words:

$\tau (n)=\sum _{d|n}1$.

Formula in terms of prime factorization

Suppose we have:

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

Then:

$\tau (n)=\prod _{i=1}^{r}(k_{i}+1)$.

Behavior

Lower bound

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

$\tau (p)=2\ \forall \ p$.

In particular:

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

Upper bound

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