Arithmetic derivative

The arithmetic derivative was introduced by Spanish mathematician Josè Mingot Shelly in 1911.[1][2] The arithmetic derivative also appeared in the 1950 Putnam Competition.[3]

For natural numbers ${\displaystyle n}$ the arithmetic derivative ${\displaystyle D(n)}$[note 1] is defined as follows:

• ${\displaystyle D(p)\;=\;1}$ for any prime ${\displaystyle p}$.
• ${\displaystyle D(pq)\;=\;D(p)q\,+\,pD(q)}$ for any ${\displaystyle p{\textrm {,}}\,q\;\in \;\mathbb {N} }$ (Leibniz rule).

Edward J. Barbeau extended it to all integers by proving that ${\displaystyle D(-x)\;=\;-D(x)}$ 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 ${\displaystyle \mathbb {Q} }$:

${\displaystyle D\left({\frac {p}{q}}\right)={\frac {D(p)q-pD(q)}{q^{2}}}\ .}$[4][5]

Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents of primes ${\displaystyle e_{i}}$ are allowed to be arbitrary rational numbers, allowing expressions like ${\displaystyle D({\sqrt {3}})}$ to be computed. [6]

The arithmetic derivative can also be extended to any unique factorization domain,[6] 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.[7]

The Leibniz rule implies that ${\displaystyle D(0)=0}$ (take ${\displaystyle p=q=0}$) and ${\displaystyle D(1)=0}$ (take ${\displaystyle p=q=1}$).

The power rule is also valid for the arithmetic derivative. For any integers p and n ≥ 0:

${\displaystyle D(p^{n})=np^{n-1}D(p).}$

This allows one to compute the derivative from the prime factorisation of an integer, ${\displaystyle x=\prod _{i=1}^{\omega (x)}{p_{i}}^{v_{p_{i}}(x)}}$:

${\displaystyle D(x)=\sum _{i=1}^{\omega (x)}\left[v_{p_{i}}(x)\left(\prod _{j=1}^{i-1}{p_{j}}^{v_{p_{j}}(x)}\right)p_{i}^{v_{p_{i}}-1}\left(\prod _{j=i+1}^{\omega (x)}{p_{j}}^{v_{p_{j}}(x)}\right)\right]=\sum _{i=1}^{\omega (x)}{\frac {v_{p_{i}}(x)}{p_{i}}}x=\sum _{\stackrel {p\mid x}{p{\text{ prime}}}}{\frac {v_{p}(x)}{p}}x}$

where ${\displaystyle \omega (x)}$, a prime omega function, is the number of distinct prime factors in ${\displaystyle x}$, and ${\displaystyle v_{p}(x)}$ is the p-adic valuation of ${\displaystyle x}$.

For example:

${\displaystyle D(60)=D(2^{2}\cdot 3\cdot 5)=\left({\frac {2}{2}}+{\frac {1}{3}}+{\frac {1}{5}}\right)\cdot 60=92,}$

or

${\displaystyle D(81)=D(3^{4})=4\cdot 3^{3}\cdot D(3)=4\cdot 27\cdot 1=108.}$

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

${\displaystyle 0,0,1,1,4,1,5,1,12,6,7,1,16,1,9,\ldots }$

The logarithmic derivative ${\displaystyle \operatorname {ld} (x)={\frac {D(x)}{x}}=\sum _{\stackrel {p\mid x}{p{\text{ prime}}}}{\frac {v_{p}(x)}{p}}}$ is a totally additive function: ${\displaystyle \operatorname {ld} (x\cdot y)=\operatorname {ld} (x)+\operatorname {ld} (y).}$

E. J. Barbeau examined bounds of the arithmetic derivative.[8] He found that

${\displaystyle D(n)\leq {\frac {n\log _{2}n}{2}}}$

and

${\displaystyle D(n)\geq \Omega (n)n^{\frac {\Omega (n)-1}{\Omega (n)}}}$

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

Dahl, Olsson and Loiko found the arithmetic derivative of natural numbers is bounded by[9]

${\displaystyle D(n)\leq {\frac {n\log _{p}n}{p}}}$

where ${\displaystyle p}$ is the least prime in ${\displaystyle n}$ and equality holds when ${\displaystyle n}$ is a power of ${\displaystyle p}$.

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.

We have

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

and

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

for any δ > 0, where

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

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 ${\displaystyle k>1}$ the existence of an ${\displaystyle n}$ so that ${\displaystyle D(n)=2k}$. The twin prime conjecture would imply that there are infinitely many ${\displaystyle k}$ for which ${\displaystyle D^{2}(k)=1}$.[6]

• Arithmetic function
• Derivation (differential algebra)
• p-derivation

• 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. arXiv:math/0508364. ISSN 1530-7638. Zbl 1065.05019.
• Dahl N., Olsson J., Loiko A., Investigation of the properties of the arithmetic derivative.
• Balzarotti, Giorgio; Lava, Paolo Pietro (2013). La derivata aritmetica. Alla scoperta di un nuovo approccio alla teoria dei numeri. Milan: Hoepli. ISBN 978-88-203-5864-8.
• Koviˇc, Jurij (2012). "The Arithmetic Derivative and Antiderivative" (PDF). Journal of Integer Sequences. 15 (3.8).
• Haukkanen, Pentti; Merikoski, Jorma K.; Mattila, Mika; Tossavainen, Timo (2017). "The arithmetic Jacobian matrix and determinant" (PDF). Journal of Integer Sequences. 20. Article 17.9.2. ISSN 1530-7638.
• Haukkanen, Pentti; Merikoski, Jorma K.; Tossavainen, Timo (2018). "The arithmetic derivative and Leibniz-additive functions". Notes on Number Theory and Discrete Mathematics. 24.
• Haukkanen, Pentti (2019). "Generalized arithmetic subderivative". Notes on Number Theory and Discrete Mathematics. 25.
• Haukkanen, Pentti; Merikoski, Jorma K.; Tossavainen, Timo (2020). "Arithmetic Subderivatives: p-adic Discontinuity and Continuity". Journal of Integer Sequences. 23. Article 20.7.3. ISSN 1530-7638.
• Haukkanen, Pentti; Merikoski, Jorma K.; Tossavainen, Timo (2020). "Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative". Mathematical Communications. 25.