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.


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.


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



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