arithmetic derivative

The arithmetic derivative was introduced by Spanish mathematician Josè Mingot Shelly in 1911.{{cite journal |last1=Shelly |first1=D. J. M. |title=Una cuestión de la teoria de los numeros |journal=Association Esp. Granada |date=1911 |pages=1–12 |jfm=42.0209.02 |url=https://zbmath.org/?q=an:42.0209.02}}{{cite book |last1=Lava |first1=Paolo Pietro |last2=Balzarotti |first2=Giorgio |title=La derivata aritmetica: Alla scoperta di un nuovo approccio alla teoria dei numeri}} The arithmetic derivative also appeared in the 1950 Putnam Competition.{{cite web |last1=Scholes |first1=John |title=10th Putnam 1950 |url=https://prase.cz/kalva/putnam/putn50.html}}

For natural numbers {{mvar|n}}, the arithmetic derivative {{math|D(n)}}In this article we use Oliver Heaviside's notation {{math|D(n)}} for the arithmetic derivative of {{mvar|n}}. There are various other notations possible, such as {{math|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 {{math|D^k}} for second and higher-order arithmetic derivatives. is defined as follows:

• {{math|1=D(p) = 1}} for any prime {{mvar|p}}.
• {{math|1=D(mn) = D(m)n + mD(n)}} for any $m,\; n\; \backslash in\; \backslash N$ (Leibniz rule).

Edward J. Barbeau extended the domain to all integers by showing that the choice {{math|1=D(−n) = −D(n)}} uniquely extends the domain to the integers and 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 $\backslash Q$:

:$D\backslash !\backslash left(\backslash frac\{m\}\{n\}\backslash right)\; =\; \backslash frac\{D(m)n-m\; D(n)\}\{n^2\}\; .${{cite journal |last1=Barbeau |first1=Edward |title=Remarks on an Arithmetic Derivative |journal=Canadian Mathematical Bulletin |year=1961 |volume=4 |issue=2 |page=117-122 |doi=10.4153/CMB-1961-013-0|doi-access=free }}{{cite journal |last1=Barbeau |first1=Edward |title=Problem |journal=Canad. Math. Congress Notes |date=April 1973 |volume=5 |issue=8 |page=6-7}}

Victor Ufnarovski and Bo Åhlander expanded it to the irrationals that can be written as the product of primes raised to arbitrary rational powers, allowing expressions like $D(\backslash sqrt\{3\}\backslash ,)$ to be computed. {{cite journal |last1=Ufnarovski |first1=Victor |last2=Ahlander |first2=Bo |title=How to Differentiate a Number |journal=Journal of Integer Sequences |date=2003 |volume=6 |issue=3 |url=https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.pdf}}

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 arithmetic derivative has also been extended to the ring of integers modulo n.{{cite journal |last1=Krebs |first1=Mike |last2=Emmons |first2=Caleb |last3=Shaheen |first3=Anthony |title=How to Differentiate an Integer Modulo n |journal=The College Mathematics Journal |date=November 2009 |volume=40 |issue=5 |pages=345–353 |doi=10.4169/074683409X475661 |s2cid=122997343 |url=https://www.tandfonline.com/doi/abs/10.4169/074683409X475661|url-access=subscription }}

The Leibniz rule implies that {{math|1=D(0) = 0}} (take {{math|1=m = n = 0}}) and {{math|1=D(1) = 0}} (take {{math|1=m = n = 1}}).

The power rule is also valid for the arithmetic derivative. For any integers {{mvar|k}} and {{math|n ≥ 0}}:

:$D(k^n)\; =\; nk^\{n-1\}\; D(k).$

This allows one to compute the derivative from the prime factorization of an integer, $x\; =\; \backslash prod\backslash limits\_\{p\; \backslash in\; \backslash mathbb\{P\}\}\; p^\{n\_p\}$ (in which $n\_p\; =\; \backslash nu\_p(x)$ is the p-adic valuation of {{mvar|x}}) :

:$D(x)\; =\; \backslash sum\backslash limits\_\{p\; \backslash in\; \backslash mathbb\{P\}\}\; \backslash frac\{x\}\{p^\{n\_p\}\}\; n\_p\; \backslash ,\; p^\{n\_p-1\}\; D(p)$

= \sum_{\stackrel{p \vert x}{p \in \mathbb{P}}} n_p \frac x p D(p)

= x \sum_{\stackrel{p \vert x}{p \in \mathbb{P}}} \frac {n_p} {p} D(p)

.

This shows that if one knows the derivative for all prime numbers, then the derivative is fully known. In fact, the family of arithmetic partial derivative $\backslash frac\; \backslash partial\; \{\backslash partial\; p\}$ relative to the prime number $p$, defined by $\backslash frac\; \backslash partial\; \{\backslash partial\; p\}(q)=0$ for all primes $q$, except for $q=p$ for which $\backslash frac\; \backslash partial\; \{\backslash partial\; p\}(p)=1$ is a basis of the space of derivatives. Note that, for this derivative, we have $\backslash frac\; \{\backslash partial\; x\}\{\backslash partial\; p\}\; =\; n\_p\; \backslash frac\; x\; p$.

Usually, one takes the derivative such that $D(p)=1$ for all primes {{mvar|p}}, so that

:$D=\backslash sum\backslash limits\_\{p\; \backslash in\; \backslash mathbb\{P\}\}\backslash frac\; \backslash partial\; \{\backslash partial\; p\}\; \backslash text\{,\; and\; \}\; D(x)=\; x\; \backslash sum\backslash limits\_\{p\; \backslash in\; \backslash mathbb\{P\}\}\; \backslash frac\; \{n\_p\}\; p$.

With this derivative, we have for example:

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

or

:$D(81)\; =\; D(3^4)\; =\; 4\backslash cdot\; 3^3\backslash cdot\; D(3)\; =\; 4\backslash cdot\; 27\backslash cdot\; 1\; =\; 108.$

And the sequence of number derivatives for {{math|1=x = 0, 1, 2, …}} begins {{OEIS|id=A003415}}:

:$0,\; 0,\; 1,\; 1,\; 4,\; 1,\; 5,\; 1,\; 12,\; 6,\; 7,\; 1,\; 16,\; 1,\; 9,\; \backslash ldots$

The logarithmic derivative $\backslash operatorname\{ld\}(x)=\backslash frac\{D(x)\}\{x\}\; =\; \backslash sum\_\{\backslash stackrel\{p\; \backslash ,\backslash mid\backslash ,\; x\}\{p\; \backslash in\; \backslash mathbb\{P\}\}\}\; \backslash frac\; \{\backslash nu\_p(x)\}\; \{p\}$ is a totally additive function: $\backslash operatorname\{ld\}(x\; \backslash cdot\; y)\; =\; \backslash operatorname\{ld\}(x)+\backslash operatorname\{ld\}(y).$

Let $p$ be a prime. The arithmetic partial derivative of $x$ with respect to $p$ is defined as $D\_p(x)=\backslash frac\; \{\backslash nu\_p(x)\}\; \{p\}\; x.$ So, the arithmetic derivative of $x$ is given as $D(x)\; =\; \backslash sum\_\{\backslash stackrel\{p\; \backslash ,\backslash mid\backslash ,\; x\}\{p\; \backslash in\; \backslash mathbb\{P\}\}\}\; D\_p(x).$

Let $S$ be a nonempty set of primes. The arithmetic subderivative of $x$ with respect to $S$ is defined as $D\_S(x)\; =\; \backslash sum\_\{\backslash stackrel\{p\; \backslash ,\backslash mid\backslash ,\; x\}\{p\; \backslash in\; S\}\}\; D\_p(x).$

If $S$ is the set of all primes, then $D\_S(x)\; =\; D(x),$ the usual arithmetic derivative.

If $S=\backslash \{p\backslash \}$, then $D\_S(x)\; =\; D\_p(x),$ the arithmetic partial derivative.

An arithmetic function $f$ is Leibniz-additive if there is a totally multiplicative function $h\_f$ such that $f(mn)\; =\; f(m)h\_f(n)+f(n)h\_f(m)$ for all positive integers $m$ and $n$. A motivation for this concept is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h\_D(n)=n$.

The function $\backslash delta$ given in Section 3.5 of the book by Sandor and Atanassov is, in fact, exactly the same as the usual arithmetic derivative $D$.

E. J. Barbeau examined bounds on the arithmetic derivativeBarbeau, 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 and found that

: $D(n)\; \backslash leq\; \backslash frac\{n\; \backslash log\_2\; n\}\{2\}$

and

:$D(n)\; \backslash geq\; \backslash Omega(n)\backslash ,\; n^\{\backslash frac\{\backslash Omega(n)-1\}\{\backslash Omega(n)\}\}$

where {{math|Ω(n)}}, a prime omega function, is the number of prime factors in {{mvar|n}}. In both bounds above, equality always occurs when {{mvar|n}} is a power of 2.

Dahl, Olsson and Loiko found the arithmetic derivative of natural numbers is bounded byDahl, 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

: $D(n)\; \backslash leq\; \backslash frac\{n\; \backslash log\_p\; n\}\{p\}$

where {{mvar|p}} is the least prime in {{mvar|n}} and equality holds when {{mvar|n}} is a power of {{mvar|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 $\backslash mathbb\{Q\}$ to $\backslash mathbb\{Q\}$).

We have

:$\backslash sum\_\{n\; \backslash le\; x\}\; \backslash frac\{D(n)\}\{n\}\; =\; T\_0\; x\; +\; O(\backslash log\; x\; \backslash log\backslash log\; x)$

and

:$\backslash sum\_\{n\; \backslash le\; x\}\; D(n)\; =\; \backslash left(\backslash frac\{1\}\{2\}\backslash right)T\_0\; x^2\; +\; O(x^\{1+\backslash delta\})$

for any δ > 0, where

:$T\_0\; =\; \backslash sum\_p\; \backslash frac\{1\}\{p(p-1)\}.$

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 {{math|k > 1}} the existence of an {{mvar|n}} so that {{math|1=D(n) = 2k}}. The twin prime conjecture would imply that there are infinitely many {{mvar|k}} for which {{math|1=D²(k) = 1}}.

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

{{reflist|group=note}}

{{reflist}}

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Category:Additive functions

Category:Arithmetic functions

Category:Number theory

Category:Generalizations of the derivative