# Does this multiplicative function have a name? If so, what is known about it?

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 $$\sum_{n \leq X} \psi(n)$$?

$$\psi$$ 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 and standard bounds that $$\sum_{n \leq X} \psi(n)\sim\frac{\zeta(2)}{2\zeta(4)}X^2=\frac{15}{2\pi^2}X^2.$$

• I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $\sum_{n\le x} \varphi_1(n)=\frac{\zeta(2)}{2\zeta(4)}x^2+O(x\log x)$. (Apostol denotes the function by $\varphi_1$.) – Martin Sleziak Oct 18 at 13:58
• In the field of automorphic forms, this function is often denotes $\nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms). – Peter Humphries Oct 18 at 14:07
• @MartinSleziak: Thanks for your comment. Clearly, one can also prove this statement by elementary means, by making use of the convolution I mentioned. – GH from MO Oct 18 at 14:35
• @PeterHumphries: Indeed the function looked familiar! It is also the index of $\Gamma_0(n)$ in $\Gamma_0(1)$ if I am not mistaken. – GH from MO Oct 18 at 14:36
• @GHfromMO, yep, which is why it comes up in the field automorphic forms so often! – Peter Humphries Oct 18 at 15:03

This is also called the Dedekind $$\psi$$ function: $$\psi(n):=n\prod_{p|n}(1+p^{-1})$$