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Number Theory

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Moderate

Definition of Arithmetic Function

NUMTH-7UI1FB

An arithmetic function is a complex-valued function $f(n)$ defined for all $n\in \mathbb{N}$, i.e. $f:\mathbb{N}\rightarrow\mathbb{C}$.
Which of the following statements are true? Check ALL that apply.

A

There is an arithmetic function $f:\mathbb{N}\rightarrow\mathbb{C}$ which is surjective.

B

There is no arithmetic function $f$ such that $\{f(n): n\in\mathbb{N}\}$ is dense in $\mathbb{C}$.

C

Any function $f:\mathbb{R}\rightarrow \mathbb{C}$ can be regarded as an arithmetic function by restricting $f$ to $\mathbb{N}$.

D

Any arithmetic function $f:\mathbb{N}\rightarrow \mathbb{C}$ can be extended to $\mathbb{R}$ in a way that $f:\mathbb{R}\rightarrow\mathbb{C}$ is a continuous function.

E

The function $f(n)=\sin n$ is not an arithmetic function.