Sum of Möbius function over Complementary Divisors of Square-Divisors

While considering the summatory function of the Liouville Function, we found that

which takes the value of 1 when n is a perfect square and 0 otherwise.  The following question was given as a challenge and we present its solution in this article.

Question
Find

Solution:-
From
     ` \sum_{d|n}\lambda(d)=\bb{1}_\text{PerfectSquare}(n) `
using the Möbius Inversion formula, we have
     ` \sum_{d|n}\mu(\frac{n}{d})\bb{1}_\text{PerfectSquare}(d)=\lambda(n) `
The summands will be zero except when ` d=D^2 `.  Hence we get
     ` \sum_{D^2|n}\mu(\frac{n}{D^2}) =\lambda(n) `
[End]

Möbius Inversion Formula

Möbius Inversion Formula (by August Ferdinand Möbius) states that if ` Df ` the summatory function of f

then

 

where ` f ` is an arithmetic function.  This is more like a "formula machine", which, when given an input ` f `, gives you a version of formula as an output (Well, as long as you know what ` Df ` is).  ` \mu ` is the Möbius function.  [ By the way, Möbius was also famous for his Möbius strip. ]  Using Dirichlet Convolutions (* operation), the formula just says that

and since the *-inverse of the ` 1(n) ` function is the Möbius function ` \mu(n) `

the formula can be proven succinctly as follows.
Proof:     ` \mu \mbox{*} Df` `
` =  \mu \mbox{*} (1 \mbox{*} f) =  (\mu \mbox{*} 1) \mbox{*} f   =  delta_1 \mbox{*} f    =    f  `
(proven)

Example
For `f(n) = \phi(n)` the Euler totient function, we know that

Hence applying Möbius Inversion we have

i.e.                                     

since ` \phi(n) = (\mu * D\phi)(n)  = \sum_{d|n}\mu(d) \cdot D\phi(\frac{n}{d})  = \sum_{d|n}\mu(d) \cdot I(\frac{n}{d}) = \sum_{d|n}\mu(d) \cdot \frac{n}{d} `

Arithmetic Functions

An arithmetic function is a function ` ZZ^+ \rightarrow S ` where `S \subseteq RR \subseteq CC `.  They try to capture the essence of various properties of positive whole numbers.

Examples
The constant ` 1 ` function maps every  n  to the number 1.

The identity function ` I ` maps every  n  to itself.

Möbius function ` \mu `

The unit function ` \delta_1 ` (or ` \epsilon ` function)

Euler's totient function ` \phi `

Number of divisors ` \tau `

Sum of divisors ` \tau `