Number Theory

Arithmetic Functions

Arithmetic Functions: Level 5 Challenges

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

f ( x y ) = 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 d 2016 μ ( d ) \large \sum_{d|2016} \mu(d) where μ denotes theMöbius functionand the d d are the positive divisors of 2016. 2016.

d n μ ( n d ) 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 n n , find f ( 2015 ) f(2015) .

Notation: μ denotes theMöbius function.

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

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

Compute

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

Notations:

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