(Redirected from Number logarithmic derivative)
There are no approved revisions of this page, so it may
not have been
reviewed.
Definition
Considering a natural number's prime factorization
where are the distinct prime factors of , ω(n) is the number of distinct prime factors of and are positive integers,
the arithmetic logarithmic derivative of is defined as
where is the arithmetic derivative of .
Arithmetic logarithmic derivative of zero
The arithmetic logarithmic derivative of a zero is undefined
- , which is undefined.
Arithmetic logarithmic derivative of units
The arithmetic logarithmic derivative of a unit is
Arithmetic logarithmic derivative of primes
The arithmetic logarithmic derivative of a prime is
Properties
Arithmetic logarithmic derivative of a product
For any nonzero integer
The arithmetic logarithmic derivative of a product has the property
or
where and are any nonzero integers.
Thus
where the are any nonzero integers.
Arithmetic logarithmic derivative of powers
Also
where is any nonzero integer and is any integer (for we get , which is the wanted result.)
Arithmetic logarithmic derivative of a quotient
The arithmetic logarithmic derivative of a quotient has the property
or
where and are any nonzero integers.
See also