I have a question about the definition of divisor functions when I was reading primes in tuples by Goldston, Pintz, and Yıldırım:
Let $\omega(q)$ denote the number of prime factors of a squarefree integer $q$. For any real number $m$, we define $d_m(q) = m^{\omega(q)}$. This agrees with the usual definition of the divisor functions when $m$ is a positive integer.
Can anyone tell me why this definition agrees with usual definition of divisor functions?
The identity $d_m(q)=m^{\omega(q)}$ holds only when $q$ is a squarefree integer. If $q=p_1\cdots p_k$ (so that $k=\omega(q)$), then there are precisely $m^k$ ordered $m$-tuples of positive integers $(d_1,\dots,d_m)$ such that $d_1\cdots d_m=q$: exactly one coordinate $d_j$ is divisible by $p_1$, exactly one is divisible by $p_2$, and so on, and these choices can be made independently.