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 $\sum\_{n \leq X} \psi(n)$?



Does this multiplicative function have a name? If so, what is known about it?
$\psi$ is the multiplicative convolution of $\mu^2$ and the identity function, hence its Dirichlet series is 
$$\sum\_{n=1}^\infty\frac{\psi(n)}{n^s}=\frac{\zeta(s)\zeta(s-1)}{\zeta(2s)},\qquad\Re(s)>2.$$ 
This implies by [Perron's formula](https://en.wikipedia.org/wiki/Perron%27s_formula) and standard bounds that 
$$\sum\_{n \leq X} \psi(n)\sim\frac{\zeta(2)}{2\zeta(4)}X^2=\frac{15}{2\pi^2}X^2.$$


