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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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Arithmetic expansion of harmonic sum
Note: I have modified the initial question as follows:
Let $w_1, w_2, \ldots, w_d$ be positive weights, and $x_1, x_2, \ldots, x_d$ be positive variables. Now, let us consider the following harmonic s …