Let $n \in \mathbb{Z}$ with $n > 0$. Let $F(n) = \sum_{d \mid n} \lambda(d)$. Prove that $$F(n) = \begin{cases}1, \quad \text{if }n \text{ is a perfect square}\\ 0, \quad \text{otherwise} \end{cases} $$
By the Fundamental Theorem of Arithmetic, all $d$'s admit a prime factorization $d=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ for primes $p_i$ and nonnegative integers $a_i$. So $\lambda(d)=\lambda(p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k})$. Now the idea is that all the divisors will cancel in pairs of $1$ and $-1$ when $n$ is a perfect square, except the divisor $1$, and so the sum will total $1$. How do I prove this rigorously? If I just choose a generic divisor and write it's prime factorization I don't find anything I can generalize. Can you help?
Best Answer
You can prove it using "multiplicative" property and it's simple but I just solved it without it and I believe that it's much more beautiful.
Consider $$n=p_{1}^{a_1} p_2^{a_2} ... p_k^{a_k}$$
So $\Omega(n)=\sum a_i$ and $\lambda(n)=(-1)^{\Omega(n)}$
Now $$\sum_{d|n}\lambda(d)=\sum_{d|n}(-1)^{\Omega(d)}$$
So it's suffices to calculate $D$; the difference of the number of divisor with even $\Omega$ and odd $\Omega.$
Now consider $f(x)=(1+x+x^2+...+x^{a_1})(1+x+x^2+...+x^{a_2})...(1+x+x^2+...+x^{a_k}) $
The coefficient of $x^r$ in above expansion is equal to the number of solutions of this equation:
$$x_1+x_2+...+x_k=r $$ $$0 \le x_i \le a_i$$ which is the number of divisors of $n$ with $\Omega$ equals to $r$.
Hence, $f(-1)=D.$
But $f(-1)$ is $0$ if at least one of $a_i$ is odd and $f(-1)=1$ if $n$ is a perfect square. $Q.E.D$
As a second solution:
It's easy to check $\lambda=1_{Sq}*\mu $
$[$ Actually the summation has only one term.$]$
Now convolve both sides with the $1$ function [which is inverse of $\mu$]
$$ \lambda * 1=1_{Sq}*\mu*1=1_{Sq} $$
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