Talk:Arithmetic derivative

Merged with p-derivation

The title of this page should be changes to Lagarias Arithmetic Derivative. There are many versions of arithmetic derivatives and it is misleading to readers that this is "the" arithmetic derivative. Equal weight should be given to Buium's (p-derivations) (phi(x) - x^p)/p and Ihara's arithmetic derivatives (teichmuller(x) - x)/p. — Preceding unsigned comment added by 169.232.212.169 (talk) 17:55, 26 September 2013 (UTC)

It would actually be better if p-derivation was moved to Arithmetic Derivative. There should also be a section on Ihara derivatives and possibly connections to Mochizuki's work.

The so called "Arithmetic derivation" and "p-derivations" are quite different generalization of derivation on rings. For the former, the additive property is dropped, and for the latter, the additive and multiplicative are modified: I do not think they should merge. AlainD (talk) 12:42, 14 December 2015 (UTC)

I think the derivatives should be changed. We can have all these different subjects in and actually while they are different the concept of AD is very similar. — Preceding unsigned comment added by Mtheorylord (talk • contribs) 23:05, 10 September 2016 (UTC)

Connexion with Field with one element

I'm prepared to believe the comment that The arithmetic derivative is the relative differential of Z over the field with one element. but would like to see that at least stated, and prefereably explained and sourced, at one of the two articles. Otherwise the link is simply baffling. Richard Pinch (talk) 05:54, 26 September 2008 (UTC)

Well, I just spent a few hours making Field with one element not quite so bad, and I looked into references for this statement. Unfortunately, I couldn't find any. :-( So at this point, I feel like there's two ways we can go. One is to put an unsourceable statement into Field with one element, and the other is to de-link this article. The first, unfortunately, violates WP policy, so it looks like we have to go with the second. The unfortunate thing is that I'm pretty sure that the arithmetic derivative really is the relative differential, so I feel like we're losing information. On the other hand, F₁ is so poorly understood that I wouldn't be surprised if that intuition were wrong, and we shouldn't be saying things that might be wrong unless we're quoting somebody. Consequently I've removed the link from the present article. Ozob (talk) 18:05, 21 November 2009 (UTC)

Feynman derivative

The quick derivative formula by Richard Feynman ("Tips on physics", p.20-22) matches the given formula for the general prime factorization,
$x'=x\sum _{i=1}^{k}e_{i}{\frac {p_{i}'}{p_{i}}}$
since $p_{i}'=1$. It is easy to prove it by induction on k. (It would look nicer with the common x taken out of the sum.) Baredodo (talk) 11:03, 21 November 2009 (UTC)

In facts, the above formula attributed to R. Feynman was known by Liebnitz, and the "arithmetioc derivative" is characterized by $p'=1$ for all primes. — Preceding unsigned comment added by AlainD (talk • contribs) 12:29, 14 December 2015 (UTC)

The constant T0

The constant T₀ is defined by Barbeau (1961) as

$T_{0}=\sum _{p\geq 2}{\frac {1}{p(p-1)}}\approx 0.749\ .$

Unfortunately a quick computation suggests that the correct value is 0.773. Can anyone suggest a resolution? Deltahedron (talk) 16:13, 19 July 2014 (UTC)

While it's not a resolution, I agree with you that the sum is incorrect. The sum over just p = 2, 3, 5, 7 exceeds 0.749. Summing over all p less than 10⁸ got me 0.773156668524163, though I trust no more than the first 7 digits. Ozob (talk) 13:08, 20 July 2014 (UTC)

I wonder whether we could find a better source. First, the correct value of T_0 is indeed 0.773156669049795... (OEIS: A136141). Second, Barbeau is being rather sloppy here, as the same kind of elementary argument as in his paper actually shows the better bounds

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

$\sum _{n\leq x}n'={\tfrac {1}{2}}T_{0}x^{2}+O(x\log x).$

—Emil J. 14:36, 6 August 2014 (UTC)

Lagarias derivative

Why is it called Lagarias derivative ? — Preceding unsigned comment added by AlainD (talk • contribs) 17:46, 18 October 2015 (UTC)

I have not found a reliable source stating this as of yet --2602:306:348E:17B0:B056:C446:4A06:57D (talk) 23:55, 23 September 2020 (UTC)

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inequalities

What is k in the first formula of this setion? — Preceding unsigned comment added by 217.87.8.122 (talk) 19:49, 23 February 2018 (UTC)

The k should be p, the prime number. I have checked that with the reference included. --Profejmpc (talk) 20:20, 12 August 2020 (UTC)

Proof of consistency/well-definedness?

The article currently defines the arithmetic derivative this way:

For natural numbers the arithmetic derivative is defined as follows:

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

It's not immediately obvious, looking at the above, that it provides a unique and consistent characterization of a function on the natural numbers. Would it make sense to include a brief proof (or sketch of a proof) that there is indeed a unique function satisfying the definition?

And, for that matter, should we rephrase the first part to make this more explicit, e.g. as:

For natural numbers the arithmetic derivative is defined as the unique function $x\mapsto x'$ that satisfies the following conditions:

?

—Ruakh_TALK 03:48, 3 December 2018 (UTC)

Corrections in formulae and references

I have made several changes:

- Inequality with k was wrong. It must be p. This formula is not from Bardeau. I have added reference. - Bardeau wrote inequality with prime 2. Added reference inside text.

I think these are acceptable changes, but I apologize if someone thinks something different.

--Profejmpc (talk) 20:39, 12 August 2020 (UTC)

References

The references could be better referencing to a concrete fragment of text inside the article, not just included as a whole. Some suggestions?

--Profejmpc (talk) 20:49, 12 August 2020 (UTC)