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# Arithmetic Functions

It might seem counterintuitive, but if you convolute these complicated things, they'll simplify in no time! Extend your understanding of the totient function by studying this class of functions.

# Arithmetic Functions: Level 5 Challenges

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 $$f(2000)$$?

Find $\large \sum_{d|2016} \mu(d)$ where $$\mu$$ denotes the Möbius function and the $$d$$ are the positive divisors of $$2016.$$

$\sum_{d|n}\mu\left(\frac{n}{d}\right)f(d)=n$

If $$f(d)$$ is an arithmetic function such that the equation above holds for all positive integers $$n$$, find $$f(2015)$$.

Notation

$\sum_{\mu(n)=1} \frac{1}{n^2} = \frac{A}{B\pi^C}$

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 $$A+B+C$$.

Compute $\large \sum_{d|2015!}\mu(d)\phi(d).$

Notations

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