TITLE: Does this multiplicative function have a name? If so, what is known about it?
QUESTION [6 upvotes]: It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. We know that $\phi(p^k) = p^{k-1} (p-1)$ for all primes $p$. 
What about the multiplicative function $\psi$ defined on the primes by $\psi(p^k) = p^{k-1} (p+1)$? Does it have a name? If so, what's known about it?
For example, can one evaluate $\sum_{n \leq X} \psi(n)$?

REPLY [8 votes]: This is also called the Dedekind $\psi$ function:
$$
\psi(n):=n\prod_{p|n}(1+p^{-1})
$$
See also A001615 and A158523.