Arithmetic derivative

In number theory, the 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.

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Definition

For natural numbers defined as follows:

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_ip_1^{e_1}\cdots p_i^{e_i-1}\cdots p_k^{e_k} = \sum_{i=1}^k \frac{e_i}{p_i} x.

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{align} 81' = (3^4)' & = (9\cdot 9)' = 9'\cdot 9 %2B 9\cdot 9' = 2[9(3\cdot 3)'] \\ & = 2[9(3'\cdot 3 %2B 3\cdot 3')] = 2[9\cdot 6] = 108 = 4\cdot 3^3. \end{align}

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

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. He 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. 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.

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 occurs only if 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