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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 $$a \perp b.$$ $\beth$ …