Arithmetic logarithmic derivative | number theory | advanced level

Episode 108.

Arithmetic logarithmic derivative | number theory | advanced level.

Branch of mathematics: number theory.
Difficulty level: advanced.

The arithmetic derivative is a function $D$ from natural numbers to natural numbers defined by the 2 properties:
1. For any prime number $p$, we have $D(p)=1$.
2. For any 2 natural numbers $m$ and $n$, we have $D(m \cdot n) = D(m) \cdot n + m \cdot D(n)$.

The arithmetic logarithmic derivative is defined as $ld(n) = \frac{D(n)}{n}$.

Theorem. For any 2 natural numbers $m$ and $n$, we have $ld(m \cdot n) = ld(m) + ld(n)$.

Theorem. If $n=\prod_{i}{p_i}$ is the expression of $n$ into the product of prime numbers (not necessarily distinct), then $ld(n) = \sum_{i}{\frac{1}{p_i}}$.

Mathematics. Number theory.
#Mathematics #NumberTheory

The same video on YouTube:
https://youtu.be/3jPy41jM7qw

The same video on Telegram:
https://t.me/mathematical_bunker/133