Divisor sum or möbius inversion of the Jacobi Symbol?

Question

We're doing number theory and we've separately covered the topics of Quadratic Reciprocity and Multiplicative Functions.

I just noticed that if one of the parameters in the Jacobi symbol is constant - i.e. either a or m in $\left(\frac{a}{m}\right)$ then it is also a multiplicative function.

Since it is a multiplicative function, so would its divisor sum and möbius inversion be. That is, the functions

$$f(m) = \sum_{d\ |\ m}\left(\frac{a}{d}\right),\ g(a) = \sum_{d\ |\ a}\left(\frac{d}{m}\right),\ h(m) = \sum_{d\ |\ m}\left(\frac{a}{d}\right)\mu(n/d), \ i(m) = \sum_{d\ |\ m}\left(\frac{d}{m}\right)\mu(n/d)$$

will all be multiplicative functions. I tried finding information on these but did not find any. Are these functions known?

Answer 1

What I would suggest you try is the following approach. Note that the Jabcobi symbol $(n / k)$ is a Dirichlet character modulo $k$ (let's denote it by the shorthand $\chi(n))$. The Dirichlet series of this character is then given by the L-function $L(s, \chi)$ for $Re(s) > 1$, say. Then the Dirichlet series for the first two functions is of the form $FG(s) = \zeta(s) L(s, \chi)$ and for the second two functions is of the form $HI(s) = L(s, \chi) / \zeta(s)$. Now in specific cases of the choice of $\chi$ (a.k.a your Jacobi symbol here) where you can evaluate the L-function involved sufficiently well (I guess this depends on your end purposes) you can apply Perron's formula to each respective Dirichlet series case to obtain an exact expression for the summatory functions of $f/g/h/i$ as $\sum_{n \leq x} f/g/h/i(n)$. Finally, to get a closed-form expression for the initial function, you can take differences with respect to $x$ in Perron's formula to arrive at the end result.