Number Theory

# 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: $$\mu$$ denotes the Möbius function.

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

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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