Divisor Functions

Suppose

where $p_1,p_2,\ldots,p_k$ are distinct primes. Every positive divisor $d$ of $n$ has a unique form

where

There are $\alpha_1+1$ possible choices for $\beta_1$, namely $0,1,\ldots,\alpha_1$. Similarly, there are $\alpha_2+1$ choices for $\beta_2$, and so on. Since these choices are independent,

Every positive divisor of $n$ has the form

where $0\le\beta_i\le\alpha_i$. Consider the product

When this product is expanded, we choose one power of each prime $p_i$. Thus every resulting term has the form

so it is a positive divisor of $n$. Conversely, every positive divisor of $n$ appears exactly once in this expansion. Therefore,

Using the finite geometric-series formula,

Hence,

Let $t$ be the number of positive divisors of $n$, and let the positive divisors be

If $d$ is a positive divisor of $n$, then $n/d$ is also a positive divisor of $n$.

Let

be the product of all positive divisors. Since the map $d\mapsto n/d$ permutes the positive divisors,

Since $D>0$,