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 (the Lagarias arithmetic derivative), such as Ihara's arithmetic derivative and Buium's arithmetic derivatives.
Extensions beyond natural numbers
Edward J. Barbeau extended the domain to all integers by showing that the choice D(−n) = −D(n), which uniquely extends the domain to the integers, 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 :
The arithmetic derivative can also be extended to any unique factorization domain (UFD), 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 Leibniz rule implies that D(0) = 0 (take m = n = 0) and D(1) = 0 (take m = n = 1).
The power rule is also valid for the arithmetic derivative. For any integers k and n ≥ 0:
This allows one to compute the derivative from the prime factorization of an integer, :
The logarithmic derivative is a totally additive function:
The arithmetic partial derivative of with respect to is defined as So, the arithmetic derivative of is given as
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 .
Inequalities and bounds
E. J. Barbeau examined bounds on the arithmetic derivative and found that
Dahl, Olsson and Loiko found the arithmetic derivative of natural numbers is bounded by
where p is the least prime in n and equality holds when n is a power of 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 (note that this means that the arithmetic derivative is not a continuous function from to ).
Order of the average
for any δ > 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 k > 1 the existence of an n so that D(n) = 2k. The twin prime conjecture would imply that there are infinitely many k for which D2(k) = 1.
- In this article we use Oliver Heaviside's notation D(n) for the arithmetic derivative of n. There are various other notations possible, such as n′; 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 D k for second and higher-order arithmetic derivatives.
- Shelly, D. J. M. (1911). "Una cuestión de la teoria de los numeros". Association Esp. Granada: 1–12.
- Lava, Paolo Pietro; Balzarotti, Giorgio. La derivata aritmetica: Alla scoperta di un nuovo approccio alla teoria dei numeri.
- Scholes, John. "10th Putnam 1950".
- Barbeau, Edward. "Remarks on an Arithmetic Derivative". Canadian Mathematical Bulletin. 4 (2): 117-122. doi:10.4153/CMB-1961-013-0.
- Barbeau, Edward (April 1973). "Problem". Canad. Math. Congress Notes. 5 (8): 6-7.
- Ufnarovski, Victor; Ahlander, Bo (2003). "How to Differentiate a Number" (PDF). Journal of Integer Sequences. 6 (3).
- Krebs, Mike; Emmons, Caleb; Shaheen, Anthony (November 2009). "How to Differentiate an Integer Modulo n". The College Mathematics Journal. 40 (5): 345–353. doi:10.4169/074683409X475661.
- Barbeau, E.J. (1961). Remarks on an arithmetic derivative. URL: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/1FD7F09AD3972692FC97BB23A21D0BD8/S0008439500050773a.pdf/remarks_on_an_arithmetic_derivative.pdf
- Dahl, N., Olsson, J., Loiko, A. (2011). Investigations on the properties of the arithmetic derivative. On page 4. URL: https://arxiv.org/pdf/1108.4762.pdf
- 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 (2016). "On Arithmetic Partial Differential Equations" (PDF). Journal of Integer Sequences. 19. 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.