Arithmetic derivative
In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.
There are many versions of "arithmetic derivatives", including the one discussed in this article, such as Ihara's arithmetic derivative and Buium's arithmetic derivatives.
Early history
The arithmetic derivative was introduced by Spanish mathematician Josè Mingot Shelly in 1911. The arithmetic derivative also appeared in the 1950 Putnam Competition.Definition
For natural numbers, the arithmetic derivative is defined as follows:- for any prime.
- for any .
Extensions beyond natural numbers
Edward J. Barbeau extended the domain to all integers by showing that the choice uniquely extends the domain to the integers and is consistent with the product formula. Barbeau also further extended it to the rational numbers, showing that the familiar quotient rule gives a well-defined derivative on :Victor Ufnarovski and Bo Åhlander expanded it to the irrationals that can be written as the product of primes raised to arbitrary rational powers, allowing expressions like to be computed.
The arithmetic derivative can also be extended to any unique factorization domain, such as the Gaussian integers and the Eisenstein integers, and its associated field of fractions. If the UFD is a polynomial ring, then the arithmetic derivative is the same as the derivation over said polynomial ring. For example, the regular derivative is the arithmetic derivative for the rings of univariate real and complex polynomial and rational functions, which can be proven using the fundamental theorem of algebra.
The arithmetic derivative has also been extended to the ring of integers modulo n.
Elementary properties
The Leibniz rule implies that and .The power rule is also valid for the arithmetic derivative. For any integers and :
This allows one to compute the derivative from the prime factorization of an integer, :
This shows that if one knows the derivative for all prime numbers, then the derivative is fully known. In fact, the family of arithmetic partial derivatives relative to the prime number, defined by for all primes, except for for which allows to write any derivative as an infinite sum. Note that, for this derivative, we have.
Usually, one takes the derivative such that for all primes, so that
With this derivative, we have for example:
or
And the sequence of number derivatives for begins :
Related functions
The logarithmic derivative is a totally additive function:Let be a prime. The arithmetic partial derivative of with respect to is defined as So, the arithmetic derivative of is given as
Let be a nonempty set of primes. The arithmetic subderivative of with respect to is defined as
If is the set of all primes, then the usual arithmetic derivative. If, then the arithmetic partial derivative.
An arithmetic function is Leibniz-additive if there is a totally multiplicative function such that for all positive integers and. A motivation for this concept is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative ; namely, is Leibniz-additive with.
The function given in Section 3.5 of the book by Sandor and Atanassov is, in fact, exactly the same as the usual arithmetic derivative.
Inequalities and bounds
E. J. Barbeau examined bounds on the arithmetic derivative and found thatand
where, a prime omega function, is the number of prime factors in. In both bounds above, equality always occurs when is a power of 2.
Dahl, Olsson and Loiko found the arithmetic derivative of natural numbers is bounded by
where is the least prime in and equality holds when is a power of.
Alexander Loiko, Jonas Olsson and Niklas Dahl found that it is impossible to find similar bounds for the arithmetic derivative extended to rational numbers by proving that between any two rational numbers there are other rationals with arbitrary large or small derivatives.
Order of the average
We haveand
for any δ > 0, where