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In [[number theory]], the '''Lagarias arithmetic derivative''', or '''number derivative''', is a function defined for [[integer]]s, based on [[prime factorization]], by analogy with the [[product rule]] for the [[derivative|derivative of a function]] that is used in [[mathematical analysis]].
 
Remark: There are many verions of "Arithmetic Derivatives", there are the ones as in this article (Lagarias Arithmetic Derivative), Ihara's Arithmetic Derivative, and Buium's Arithmetic Derivatives.  
==Definition==
For [[natural numbers]] the arithmetic derivative is defined as follows:
 
* <math>p' \;=\; 1 \!</math> for any prime <math>p \!</math>.
* <math>(ab)'\;=\;a'b\,+\,ab' \!</math> for any <math>a \textrm{,}\, b \;\in\; \mathbb{N}</math> ([[product rule|Leibniz rule]]).
 
To coincide with the Leibniz rule <math>1'</math> is defined to be <math>0</math>, as is <math>0'</math>. Explicitly, assume that
 
:<math>x = p_1^{e_1}\cdots p_k^{e_k}\textrm{,}</math>
 
where <math>p_1,\, \dots,\, p_k</math> are distinct primes and <math>e_1,\, \dots,\, e_k</math> are positive integers.  Then
 
:<math>x' = \sum_{i=1}^k e_ip_1^{e_1}\cdots p_{i-1}^{e_{i-1}}p_i^{e_i-1}p_{i+1}^{e_{i+1}}\cdots p_k^{e_k} = \sum_{i=1}^k e_i\frac{x}{p_i}.</math>
 
The arithmetic derivative also preserves the power rule (for primes):
 
:<math>(p^a)' = ap^{a-1}\textrm{,}\!</math>
 
where <math>p</math> is prime and <math>a</math> is a positive integer. For example,
 
: <math>
\begin{align}
81' = (3^4)' & = (9\cdot 9)' = 9'\cdot 9 + 9\cdot 9' = 2[9(3\cdot 3)'] \\
& = 2[9(3'\cdot 3 + 3\cdot 3')] = 2[9\cdot 6] = 108 = 4\cdot 3^3.
\end{align}
</math>
 
The sequence of number derivatives for ''k'' = 0, 1, 2, ... begins {{OEIS|id=A003415}}: (This reference confuses Buium derivatives and Lagarias derivatives which are not the same thing!)
 
: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 (this may not be True, I believe there is an earlier paper of Lagarias which examines this operation). He extended it to all integers by proving that <math>(-x)' \;=\; -(x')</math> 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''':
 
:<math>\left(\frac{a}{b}\right)' = \frac{a'b-b'a}{b^2} \ .</math>
 
[[Victor Ufnarovski]] and [[Bo Åhlander]] expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents <math>e_i</math> are allowed to be arbitrary rational numbers.
 
The ''logarithmic derivative'' <math>\frac{n'}{n}</math> is a [[totally additive function]].
 
==Average order==
We have
 
:<math> \sum_{n \le x} \frac{n'}{n} = T_0 x + O(\log x \log\log x) </math>
 
and
 
:<math> \sum_{n \le x} n' = (1/2)T_0 x^2 + O(x^{1+\delta}) </math>
 
for any δ>0, where
 
:<math>T_0 = \sum_p \frac{1}{p(p-2)} \approx 0.749 \ . </math>
 
==Inequalities and bounds==
E. J. Barbeau examined bounds of the arithmetic derivative. He found that the arithmetic derivative of natural numbers is bounded by
: <math>
n' \leq \frac{n \log_k n}{k}
</math>
where ''k'' is the least prime in ''n'' and
 
: <math>
n' \geq sn^{\frac{s-1}{s}}
</math>
where ''s'' is the number of prime factors in ''n''.
In both bounds above, equality occurs only if ''n'' is a perfect power of 2, that is <math>n=2^m</math> for some ''m''.
 
[[Alexander Loiko]], [[Jonas Ernst Olsson|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.
 
==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 ''k''&nbsp;>&nbsp;1 the existence of an ''n'' so that ''n''<nowiki>'</nowiki> = 2''k''. The twin prime conjecture would imply that there are infinitely many ''k'' for which ''k''<nowiki>''</nowiki> = 1.
 
==References==
{{reflist}}
* {{cite journal | first=E. J. | last=Barbeau | title=Remarks on an arithmetic derivative | journal=[[Canadian Mathematical Bulletin]] | volume=4 | year=1961 | pages=117–122 | doi=10.4153/CMB-1961-013-0 | zbl=0101.03702 }}
* {{cite journal | first1=Victor | last1=Ufnarovski | first2=Bo | last2=Åhlander | url=http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.html | title=How to Differentiate a Number | journal=[[Journal of Integer Sequences]] | volume=6 | year=2003 | at=Article 03.3.4 | zbl=1142.11305 | issn=1530-7638 }}
* [http://planetmath.org/encyclopedia/ArithmeticDerivative.html Arithmetic Derivative]'', [[Planet Math]]'', accessed 04:15, 9 April 2008 (UTC)
* L. Westrick. ''[http://web.archive.org/web/20050426071741/http://web.mit.edu/lwest/www/intmain.pdf Investigations of the Number Derivative]''.
* Peterson, I. ''[http://www.maa.org/mathland/mathtrek_03_22_04.html Math Trek: Deriving the Structure of Numbers]''.
* {{cite journal | first=Michael | last=Stay | journal=[[Journal of Integer Sequences]] | volume=8 | year=2005 | at=Article 05.1.4 | title=Generalized Number Derivatives | url=https://cs.uwaterloo.ca/journals/JIS/VOL8/Stay/stay44.html | zbl=1065.05019 | issn=1530-7638 }}
* Dahl N., Olsson J., Loiko A., ''[http://arxiv.org/abs/1108.4762 Investigation of the properties of the arithmetic derivative]''.
 
[[Category:Number theory]]
[[Category:Generalizations of the derivative]]

Latest revision as of 11:26, 16 December 2014

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