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			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
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