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.

Remark: There are many versions 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:

$p'\;=\;1\!$ for any prime $p\!$ .
$(ab)'\;=\;a'b\,+\,ab'\!$ for any $a{\textrm {,}}\,b\;\in \;\mathbb {N}$ (Leibniz rule).

To coincide with the Leibniz rule $1'$ is defined to be $0$ , as is $0'$ . Explicitly, assume that

x=p_{1}^{e_{1}}\cdots p_{k}^{e_{k}}{\textrm {,}}

where $p_{1},\,\dots ,\,p_{k}$ are distinct primes and $e_{1},\,\dots ,\,e_{k}$ are positive integers. Then

x'=\sum _{i=1}^{k}e_{i}p_{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}}}=x\sum _{i=1}^{k}{\frac {e_{i}}{p_{i}}}

The arithmetic derivative also preserves the power rule (for primes):

(p^{a})'=ap^{a-1}{\textrm {,}}\!

where $p$ is prime and $a$ is a positive integer. For example,

{\begin{aligned}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{aligned}}

The sequence of number derivatives for k = 0, 1, 2, ... begins (sequence A003415 in the OEIS):

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, ....

E. J. Barbeau was most likely the first person to formalize this definition. He also extended it to all integers by proving that $(-x)'\;=\;-(x')$ 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:

\left({\frac {a}{b}}\right)'={\frac {a'b-b'a}{b^{2}}}\ .

Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents $e_{i}$ are allowed to be arbitrary rational numbers.

The logarithmic derivative ${\frac {x'}{x}}$ is a totally additive function.

Average order

We have

\sum _{n\leq x}{\frac {n'}{n}}=T_{0}x+O(\log x\log \log x)

and

\sum _{n\leq x}n'=(1/2)T_{0}x^{2}+O(x^{1+\delta })

for any δ>0, where

T_{0}=\sum _{p}{\frac {1}{p(p-1)}}.

Inequalities and bounds

E. J. Barbeau examined bounds of the arithmetic derivative. He found that the arithmetic derivative of natural numbers is bounded by

n'\leq {\frac {n\log _{k}n}{k}}

where k is the least prime in n and

n'\geq sn^{\frac {s-1}{s}}

where s is the number of prime factors in n. In both bounds above, equality always occurs when n is a perfect power of 2, that is $n=2^{m}$ for some m.

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.

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 > 1 the existence of an n so that n' = 2k. The twin prime conjecture would imply that there are infinitely many k for which k'' = 1.

References

Barbeau, E. J. (1961). "Remarks on an arithmetic derivative". Canadian Mathematical Bulletin. 4: 117–122. doi:10.4153/CMB-1961-013-0. Zbl 0101.03702.
Ufnarovski, Victor; Åhlander, Bo (2003). "How to Differentiate a Number". Journal of Integer Sequences. 6. Article 03.3.4. ISSN 1530-7638. Zbl 1142.11305.
Arithmetic Derivative, Planet Math, accessed 04:15, 9 April 2008 (UTC)
L. Westrick (2003). Investigations of the Number Derivative.
Peterson, I. Math Trek: Deriving the Structure of Numbers.
Stay, Michael (2005). "Generalized Number Derivatives". Journal of Integer Sequences. 8. Article 05.1.4. ISSN 1530-7638. Zbl 1065.05019.
Dahl N., Olsson J., Loiko A., Investigation of the properties of the arithmetic derivative.

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