Get Arithmetic Derivative essential facts below. View Videos or join the Arithmetic Derivative discussion. Add Arithmetic Derivative to your PopFlock.com topic list for future reference or share this resource on social media.
There are many versions of "arithmetic derivatives", including the one discussed in this article (the Lagarias arithmetic derivative), such as Ihara's arithmetic derivative and Buium's arithmetic derivatives.
The arithmetic derivative was introduced by Spanish mathematician Josè Mingot Shelly in 1911. The arithmetic derivative also appeared in the 1950 Putnam Competition.
Edward J. Barbeau 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 :
Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents of primes are allowed to be arbitrary rational numbers, allowing expressions like to be computed. 
E. J. Barbeau examined bounds of the arithmetic derivative. He found that
where , a prime omega function, is the number of prime factors in .
In both bounds above, equality always occurs when is a perfect power of 2, that is for some .
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
for any ? > 0, where
Relevance to number theory
Victor Ufnarovski and Bo Åhlander have detailed the function's connection to famous number-theoretic conjectures like the twin prime conjecture, the prime triples conjecture, and Goldbach's conjecture. For example, Goldbach's conjecture would imply, for each the existence of an so that . The twin prime conjecture would imply that there are infinitely many for which .
^In this article we use Oliver Heaviside's notation for the arithmetic derivative of . There are various other notations possible, such as ; a full discussion is available here for general differential operators, of which the arithmetic derivative can be considered one. Heaviside's notation is used here because it highlights the fact that the arithmetic derivative is a function over the integers and yields itself better notation-wise to function iteration for second and higher-order arithmetic derivatives.