Dirichlet product of arithmetic functions $\tau$ and $\sigma$

332 Views Asked by At

Let $\tau$ and $\sigma$ be the arithmetic functions defined by

  • $\tau(n)$ is the number of divisors of n, i.e $\tau(n) = \sum_{d\mid n}1$,

  • $\sigma(n)$ is the sum of divisors of n , i.e $\sigma (n) = \sum_{d\mid n}d$,

for all n.

What is $(\sigma*\tau)(12)$? (The Dirichlet product)

I know this gives $\sum_{d\mid 12}\sigma(d)\tau(12/d)$ but do not know where to go from here.