Motivation for the Derivative of an Arithmetic Function

Question

I'm currently reading through Apostol's Introduction to Analytic Number Theory. In Chapter 2, he discusses arithmetic functions (i.e. functions $f: \mathbb{N} \to \mathbb{C}$) under the Dirichlet Convolution $$ (f \ast g)(n) = \sum_{d \mid n} f(d)g(\frac{n}{d}).$$ He defines the derivative of an arithmetic function by setting $f'(n) = f(n)\log(n)$, goes on to prove that it has "nice" properties, e.g. linearity and the product rule. What I'm struggling to understand is why we chose this definition. Is there a motivation behind this definition, or are we making it because it happens to give us the structure we want?

Answer 1

Consider the following Dirichlet series:

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$$F(s)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N a(n)\ n^{-s}\right)\tag{1}$$

$$G(s)=-F'(s)=\underset{N\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^N a(n)\ \log(n)\ n^{-s}\right)\tag{2}$$

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For $a(n)=1$, $F(s)=\zeta(s)$ and $G(s)=-F'(s)=-\zeta'(s)$

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As another example, for $a(n)=\left(\left\{\begin{array}{cc} 0 & n=1 \\ \frac{\Lambda (n)}{\log (n)} & n>1 \\ \end{array}\right.\right)$, $F(s)=\log\zeta (s)$ and $G(s)=-F'(s)=-\frac{\partial\ \log\zeta(s)}{\partial s}=-\frac{\zeta'(s)}{\zeta(s)}$.