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Arithmetic logarithmic derivative

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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