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\title{Math 412  Homework 6}
\date{Due date: Oct 9, 2015}
\author{your name}


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\maketitle

Solve the following problems. Please remember to use complete sentences
and good grammar.  



\begin{enumerate}

\item (4 points) Show that if $n$ is a positive integer, then $(\sum_{d|n} \tau(d))^2=\sum_{d|n} \tau(d)^3$. 

\item (4 points) Show that if $f(n)=\sum_{d|n} \mu(d)F(n/d)$, then $F(n)=\sum_{d|n} f(d)$. 

\item (4 points) The Dirichlet product of two arithmetic functions $f$ and $g$ is defined to $(f*g)(n)=\sum_{d|n} f(d)g(n/d)$.  Show that if $f$ and $g$ are multiplicative functions, then $f*g$ is also multiplicative.  

\item (4 points) Show that $\sum_{d|n} \mu^2(d)=2^{\omega(n)}$, where $\omega(n)$ is the number of distinct prime factors of $n$. 

\item (4 points) Show that if $n$ is a positive integer, then $\phi(n)=n\sum_{d|n}\mu(d)/d$. 

\item A positive integer $n$ is called super perfect if $\sigma(\sigma(n))=2n$.  Show that \\
(a) (2 points) if $n=2^q$, where $2^{q+1}-1$ is prime, then $n$ is superperfect. \\
(b) (4 points) every even superperfect number is of form $n=2^q$, where $2^{q+1}-1$ is prime.  

\item (Bonus, 4 points) Show that if $n=p^2$, where $p$ is an odd prime, then $n$ is not superperfect. 







\end{enumerate}

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