Definition:Divisor Counting Function

Definition
Let $n$ be an integer such that $n \ge 1$.

The divisor counting function is defined on $n$ as being the total number of positive integer divisors of $n$.

It is denoted on as $\sigma_0$ (the Greek letter sigma).

That is:
 * $\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$

where $\ds \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.

Also known as
Some sources refer to this as the divisor function and denote it $\map d n$.

However, as this function is an instance of a more general definition of the divisor function, the more precise name divisor counting function is preferred.

It is also often referred to as the $\tau$ (tau) function, but there are a number of functions with such a name.

Some sources use $\nu$, but again, that also has multiple uses.

Hence the unwieldy, but practical, divisor counting function, which is non-standard.

Also see

 * Definition:Divisor Function
 * Definition:Divisor Sum Function