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\begin{document}
\noindent \textbf{Math 465 - Spring 2021} \hfill  March 25th, 2021\\
\textbf{ Group Discussion 6} \\
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\noindent \small \textit{When people tell you something's wrong or doesn't work for them, they are almost always right. When they tell you exactly what they think is wrong and how to fix it, they are almost always wrong.}\\\vspace{1mm} \hfill---
Neil Gaiman


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\begin{enumerate}

\item Prove that $\mu(n)$ is multiplicative.  

\item The Liouville $\lambda$-function is defined by $\lambda(1)=1$ and $\lambda(p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}) = (-1)^{k_1+k_2+\cdots k_r}$. 
\begin{enumerate}
\item Show that $\lambda$ is totally multiplicative.
\item Show that \[\sum_{d|n} \lambda(d) = \begin{cases} 1  & n=m^2 \\ 0 & \textbf{otherwise} \end{cases}\]
\end{enumerate}
\item Suppose $f(n)$ is multiplicative and $g(n) = \sum_{d|n} f(d)$.  Prove that $g(n)$ is multiplicative.  Hint: the goal is to show that $g(mn) = g(m)g(n)$.  Foil out the expression $\left(\sum_{d|m} f(d) \right)\left(\sum_{e|n} f(e)\right)$.
\end{enumerate}
\end{document}
