Question 2
For a positive integer $N$, sum $(-1)^{(n-1)/2}$ over all odd positive divisors $n$ of $N$, and call the sum $f(N)$.
(a) Prove, for an integer $r$ and a prime $p$, that $f(2)=1$, $f(2^r)=1$, and that $f(p)=2$ if $p=4k+1$, and $f(p)=0$ if $p=4k-1$. Prove also
$$f(p^r)=\begin{cases}1+r,&p=4k+1,\\1,&p=4k-1\text{ and }r\text{ even},\\0,&p=4k-1\text{ and }r\text{ odd}.\end{cases}$$
(b) Prove multiplicativity: if $M,N$ are positive integers and $\gcd(M,N)=1$, then $f(MN)=f(M)f(N)$. Calculate $f(5^4\cdot11^{28}\cdot17^{19})$ and $f(1980)$ and state a general rule for $f(N)$.
(a) Prove, for an integer $r$ and a prime $p$, that $f(2)=1$, $f(2^r)=1$, and that $f(p)=2$ if $p=4k+1$, and $f(p)=0$ if $p=4k-1$. Prove also
$$f(p^r)=\begin{cases}1+r,&p=4k+1,\\1,&p=4k-1\text{ and }r\text{ even},\\0,&p=4k-1\text{ and }r\text{ odd}.\end{cases}$$
(b) Prove multiplicativity: if $M,N$ are positive integers and $\gcd(M,N)=1$, then $f(MN)=f(M)f(N)$. Calculate $f(5^4\cdot11^{28}\cdot17^{19})$ and $f(1980)$ and state a general rule for $f(N)$.