Definition:Logarithmic Mean Value

Definition
Let $f: \N \to \C$ be an arithmetic function.

The logarithmic mean value of $f$ is the limit:
 * $L \left({f}\right) = \displaystyle \lim_{x \mathop \to \infty} \frac 1 {\ln x} \sum_{n \mathop \le x} \frac {f \left({n}\right)} n$

if it exists.

Also see

 * Definition:Ordinary Mean Value
 * Existence of Ordinary Mean Value Implies Existence of Logarithmic Mean Value