TITLE: Question about terminology in number theory
QUESTION [5 upvotes]: The following transformation appears often in number theory: $$F(x) = \sum_{n \le x} f \left( \frac{x}{n} \right)$$
What is the name of this transformation?
PS. I will accept as answer something that when I insert it in Google, I end up with articles about this sum.
PS2. As pointed out in the answers below, this is indeed called generalized Dirichlet convolution, see section 2.14 in "Introduction to Analytic Number Theory" By Apostol

REPLY [4 votes]: I think the standard terminology is the (generalized) Dirichlet convolution
$$
(\alpha\circ F)(x)=\sum_{n\le x} \alpha(n)F\left(\frac{x}{n}\right),
$$
where $\alpha$ is an arithmetic function, and $F$ is a real or complex valued function such that $F(x)=0$ on $0<x<1$. If $F(x)=0$ for all non-integral $x$, then this product is just the Dirichlet convolution $\alpha\ast F$. So your transformation is $x\mapsto (id \circ F)(x)$.