Definition
For natural numbers the arithmetic derivative is defined as follows:
- for any prime .
- for any (Leibniz rule).
To coincide with the Leibniz rule is defined to be, as is . Explicitly, assume that
where are distinct primes and are positive integers. Then
The arithmetic derivative also preserves the power rule (for primes):
where is prime and is a positive integer. For example,
The sequence of number derivatives for k = 0, 1, 2, ... begins (sequence A003415 in OEIS):
- 0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, ....
E. J. Barbeau was the first to formalize this definition. He extended it to all integers by proving that uniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers,showing that the familiar quotient rule gives a well-defined derivative on Q:
Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents are allowed to be arbitrary rational numbers.
The logarithmic derivative is a totally additive function.
Read more about this topic: Arithmetic Derivative
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