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An arithmetic function is one whose domain is the positive integers and whose range is a subset of the complex numbers. There are a number of important number-theoretic examples.
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Representation of a number as a sum of co-prime numbers
Conventions
We will call the following product the canonical expansion of $a$:
$$a = \prod\limits_{p_i\in\mathbb{P}}p_i^{\alpha_i}.$$
If $a$ and $b$ are co-prime we write
$$a \perp b.$$
$\beth$ …