Problem with the Möbius Inversion Formula

Question

(Edit) Can someone explain where the mistake is and what is wrong with the following "proof"?:

(i) $\sum_{d\mid n}(\sigma_k(d))$ = $\sum_{d|n}(\tau(\frac{n}{d})d^k)$ (True)

(ii) Define a function $S(n)$ such that $\sum_{d\mid n}(\sigma_k(d)) = S(n) = \sum_{d|n}(\tau(\frac{n}{d})d^k)$

(iii) The möbius inversion formula as it is stated in [Apostal, Analytic Number Theory, pg 32] says: If $f(n) = \sum_{d|n}g(d)$, then $g(n) = \sum_{d|n}f(d)\mu(\frac{n}{d})$

(iv)(a) $S(n) = \sum_{d\mid n}(\sigma_k(d))$, so we have $\sigma_k(n) = \sum_{d|n}S(d)\mu(\frac{n}{d})$

(iv)(b) $S(n) = \sum_{d\mid n}(\tau(\frac{n}{d})d^k))$, so we have $\tau(\frac{n}{n})n^k = n^k = \sum_{d|n}S(d)\mu(\frac{n}{d})$

(v) Therefore $\sigma_k(n) = n^k$

If anyone can tell me which line(s) is wrong and why it's wrong, that would be incredibly helpful. Thanks.

Answer 1

The mistake is in (iv)(b). In $$\sum_{d \mid n} \Biggl(\tau\biggl(\frac{n}{d}\biggr)d^k\Biggr) \tag{1}$$ the expression we sum over is not of the form $f(d)$, as it would need to be to apply the Möbius inversion formula (iii). The summands in $(1)$ depend on $n$ as well as on $d$, we have the form $$S(n) = \sum_{d \mid n} f(n,d)\,.$$

If we write things using the Dirichlet convolution using $u \colon n \mapsto 1$ and $\nu_k \colon n \mapsto n^k$ (thus we could write $\nu_0$ instead of $u$, but $u$ is a widely used name for that function), then the Möbius inversion formula can succinctly be stated as $$g = u \ast f \iff f = \mu \ast g\,. \tag{2}$$ The left hand side of (i) gives you $S$ in the form $u \ast \sigma_k$, precisely what $(2)$ is about, thus $\sigma_k = \mu \ast S$, or written in long form $$\sigma_k(n) = \sum_{d \mid k} \mu\biggl(\frac{n}{d}\biggr)S(d)\,,$$ by the Möbius inversion formula.

The right hand side of (i) however gives you $S$ in the form $S = \tau \ast \nu_k$, and $(2)$ says nothing about that form. A more general form of $(2)$ applies to that form however, and gives $\nu_k = \tau^{-1} \ast S$, or in long form $$n^k = \sum_{d \mid n} \tau^{-1}\biggl(\frac{n}{d}\biggr)S(d)\,,$$ where $\tau^{-1}$ is the Dirichlet inverse of $\tau$. (Since $\tau = u\ast u$ and $\mu$ is the Dirichlet inverse of $u$ we have $\tau^{-1} = \mu \ast \mu$.)

Answer 2

Let $f_k(n)=n^k$, then $\sigma_k=1* f_k$, thus $1*\sigma_k=1*(1*f_k)=(1*1)*f_k=\tau*f_k$. If you use Möbius inversion formula to $\sigma_k=1*f_k$, you get $f_k=\mu*\sigma_k$, not $f_k=\sigma_k$.