Number Theory

Arithmetic Functions

Arithmetic Functions: Level 5 Challenges

         

If f:NN f: \mathbb{N}\mapsto \mathbb{N} is a bijective function that satisfies

f(xy)=f(x)f(y) f(xy ) = f(x) f(y)

and f(2015)=42 f(2015) = 42 , what is the minimum value of f(2000) f(2000) ?

Find d2016μ(d)\large \sum_{d|2016} \mu(d) where μ\mu denotes the Möbius function and the dd are the positive divisors of 2016.2016.

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

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

Notation: μ\mu denotes the Möbius function.

μ(n)=11n2=ABπC\sum_{\mu(n)=1} \frac{1}{n^2} = \frac{A}{B\pi^C}

Let μ(n)\mu(n) denote the möbius function, the sum is taken over all positive integers nn such that μ(n)=1\mu(n)=1, with coprime positive integers AA and B.B. Find A+B+CA+B+C.

Compute

d2015!μ(d)ϕ(d). \large \sum_{d|2015!}\mu(d)\phi(d).

Notations:

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