Arithmetic Functions
If \( f: \mathbb{N}\mapsto \mathbb{N} \) is a bijective function that satisfies
\[ f(xy ) = f(x) f(y) \]
and \( f(2015) = 42 \), what is the minimum value of \( f(2000) \)?
by
Calvin Lin
Find \[\large \sum_{d|2016} \mu(d)\] where \(\mu\) denotes the Möbius function and the \(d\) are the positive divisors of \(2016.\)
by
Guillermo Templado
\[\sum_{d|n}\mu\left(\frac{n}{d}\right)f(d)=n\]
If \(f(d)\) is an arithmetic function such that the equation above holds for all positive integers \(n\), find \(f(2015)\).
Notation: \(\mu\) denotes the Möbius function.
by
Otto Bretscher
\[\sum_{\mu(n)=1} \frac{1}{n^2} = \frac{A}{B\pi^C}\]
Let \(\mu(n)\) denote the möbius function, the sum is taken over all positive integers \(n\) such that \(\mu(n)=1\), with coprime positive integers \(A\) and \(B.\) Find \(A+B+C\).
by
Julian Poon
Compute
\[ \large \sum_{d|2015!}\mu(d)\phi(d).\]
Notations:
- \(\mu(d)\) denotes the Möbius function.
- \(\phi(d)\) denotes Euler's totient function.
- \(d\) are the positive factors of \(2015!.\)
by
Otto Bretscher