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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 4

The Möbius <function> is
$$
\mu(n)=\begin{cases}
1,&n=1,\\
(-1)^r,&n\text{ is a product of }r\text{ distinct primes},\\
0,&n\text{ is divisible by a square}.
\end{cases}
$$
Möbius inversion says that
$$
g(n)=\sum_{d\mid n}f(d)
\quad\Longleftrightarrow\quad
f(n)=\sum_{d\mid n}\mu(d)g(n/d).
$$
Now
$$
\sum_{d^k\mid n}\mu(d)
$$
is the sum of $(-1)^{|S|}$ over subsets of the primes whose $k$th powers divide $n$. It is one if there are no such primes and zero otherwise, proving the power-free criterion.

Let
$$
c_n=\sum_{\substack{1\leq d\leq n\\(d,n)=1}}e^{2\pi i d/n}.
$$
Grouping all $n$th roots of unity by their exact order gives
$$
\sum_{m\mid n}c_m=\begin{cases}1,&n=1,\\0,&n>1.
\end{cases}
$$
Möbius inversion therefore yields $c_n=\mu(n)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 1F / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 1F / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 1F / Solution

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