Talk:Arithmetic function

Error in formula

under Notaion: second sentence, after "the sum or product is over all prime powers with strictly positive exponent (so 1 is not counted)::, formula is missing something.--GangofOne (talk) 03:57, 24 May 2014 (UTC)

f(6) is omitted because 6 is not a prime power. Gandalf61 (talk) 05:44, 24 May 2014 (UTC)

what I mean is, it says: sigma f(p) , shouldn't that be sigma f(p^k) .... ? GangofOne (talk) 02:04, 25 May 2014 (UTC)

I take that as a "yes". Fixed. GangofOne (talk) 03:29, 28 May 2014 (UTC)

Confused

In Arithmetic function#Miscellaneous, in the third to fifth lines, on my computer, the passage is very confused. — Preceding unsigned comment added by 38.117.79.31 (talk) 11:28, 10 July 2014 (UTC)

Yes, I modified a bit this strange layout. But the layout of the whole section is still very unconventional. Sapphorain (talk) 12:39, 10 July 2014 (UTC)

Can anyone make sense of this diagram from the article?

I have tried making sense of this diagram on two occasions, with no luck. I am considering deleting it. I looked at the history of the user who added it (three years ago) and he had a history of putting original research in articles and was eventually permanently blocked, so I tend to think this is trying to show something from a pet project, rather than some fundamental relationship of BigOmega.

The chaotic course of Ω(n) through the natural numbers (OEIS: A001222): Beginning on the height of the red line the two least significant binary digits of Ω(n) of all positive odd n below 1200 are represented by a line up (digit 1) or a line down (digit 0). The additional replacement of the “↗↘” of the semiprimes without prime factors below 5 by only one line down (here blue) even almost brings a balance between the ups and downs. The prime numbers are marked orange: orange lines for the Gaussian primes and for the other primes p additional the number x in orange so that p = x² + y² = (x + yi) (x – yi) = –i (x + yi) (y + xi) and x < y for natural x and y (OEIS: A002331).

The caption seems to contradict itself multiple times. To be specific, it first refers to the last two binary digits of Ω but then only has two choices, 0 and 1. It then departs from the stated rule by arbitrarily omitting certain primes (and when did this become about the values of the primes instead of just their count?), and then later departs the integers to refer to Gaussian primes, then loses me completely with some odd grammar.

Maybe I'm being a bit harsh; I was actually looking for the distribution of values of Ω mod 2 and at first glance this appears to be it, but reading the full description I don't trust it at all to be what I think it is. Walt (talk) 22:01, 7 September 2016 (UTC)

I agree that what is written there does not seem decipherable. --JBL (talk) 00:15, 8 September 2016 (UTC)

What are called here contradictions are actually well defined exceptions so that they are simply rule extensions. Walt then sees a problem in 2 binary digits which are of course each (could be added there) represented by a line up (for 1) or down (for 0). It also does not omit certain primes but semiprimes. These two least significant binary digits of Ω of course directly correspond to the values of Ω mod 4 (instead of 2). These values are simply balances here towards the red line by adjusting those very certain semiprimes. Can you tell which grammar is "odd" in your opinion, Walt a.k.a. User:Wroscel? --LKreissig (talk) 23:00, 24 September 2016 (UTC)

This comment is also not decipherable. Is this image your own independent work, or is it derived from some published source? --JBL (talk) 23:19, 24 September 2016 (UTC)

This image only exclusively lists the values of Ω(n) with odd n. Wikipedia contains many calculation result lists to demonstrate behaviors of functions without giving sources for each and a good encyclopedia also gives graphic representations of them instead of only pure numeric ones. If you have problems with certain word uses or text passages here then please name them concretely for improvement (e.g. should be added that "i" is the imaginary unit?) before I restore this diagram in the article. --LKreissig (talk) 11:14, 25 September 2016 (UTC)

I am a number theorist. Your diagram and the comments you wrote about it don't make any sense to me. I think this diagram can't help anybody to understand better the Omega function, except maybe yourself, and that it doesn't fit in a wikipedia article. Sapphorain (talk) 13:43, 25 September 2016 (UTC)

If the calculation being represented by this figure falls under WP:CALC then you should be able to describe, in one or two sentences, exactly what is being calculated. Neither the caption nor your comments do this. Until this changes, I agree with Sapphorain. --JBL (talk) 15:30, 25 September 2016 (UTC)

Generally the interest for mathematics as a studying subject profits from graphic demonstrations of complex mathematical circumstances so I am actually very sure that the diagram's 3 sentences, User:Joel B. Lewis, describe pretty exactly a certain binary behavior of Ω. It demonstrates that small values of Ω are so dominant that for a least balancing against Ω=3 (binary: 11) e.g. the occurrences of Ω=2 without divisor 3 could appear like one binary digit 0. Such a view to distributions of Ω values would be at present completely absent in Wikipedia without that diagram? — For a graphic contribution about that distribution, which also only lists plain calculation results, there is no need for any "I'm a great scientist but it just makes no sense to me" runs here and we can stay exactly and focused at the topic, with the diagram in the article, ok? --LKreissig (talk) 22:41, 25 September 2016 (UTC)

Precise definition

There are a few references in the article, e.g. "prime counting functions, which are not arithmetic functions", that imply that arithmetic function should have some precise definition. But the only definition I see of the term is "expresses some arithmetic property of n" which is not precise at all (at least without a link to arithmetic). Is it even a precise term, and if so, what is its definition? Can we make it clearer what the actual definition is? — Preceding unsigned comment added by Luqui (talk • contribs) 23:50, 7 June 2017 (UTC)

The apparently vague definition given by Hardy and Wright and mentioned in the first line of the article becomes clear in view of the classical functions they consider in their chapter 16. A more precise and less restrictive definition is however widely accepted by number theorists:

"An arithmetic function is any real- or complex-valued function defined on the set N of positive integers. (In other words, an arithmetic function is just a sequence of real or complex numbers, though this point of view is not particularly useful)", see [1].Sapphorain (talk) 08:15, 8 June 2017 (UTC)

... I added three classical references giving this less restrictive definition, and modified the lead accordingly. Sapphorain (talk) 15:41, 8 June 2017 (UTC)

I am sort of unhappy with this change. The previous version suffered from being not a precise definition but still morally accurate: the point of the name "arithmetic function" is that it expresses something about arithmetic, of interest to number theorists. The new definition does actually define something, but it is morally wrong: no one is really interested in the collection of all sequences of complex numbers, the vast majority of which have no meaning whatsoever for number theory (or for anything else). Sapphorain, do the sources you've added have more discussion of this definition, along the lines of the quote in your first comment, that could be used to explain that the cultural/historical background is important in the use of the name (not just a formal definition)? --JBL (talk) 20:44, 8 June 2017 (UTC)

With the more simple and precise definition the set of arithmetical function with the operations of addition and Dirichlet convolution is a unitary ring. This is proved in the Bateman-Diamond, Niven-Zuckerman, Tenenbaum books, and in most introductory courses in analytic number theory. On the other hand Hardy and Wright don't clearly mention this (although it can be inferred from their chapter 17). Sapphorain (talk) 21:33, 8 June 2017 (UTC)

My textbooks say nothing about the range or the arithmetical properties in the definition. I propose removing both clauses.—Anita5192 (talk) 22:04, 8 June 2017 (UTC)

That is a vague assertion. Who wrote your textbooks, and what definition do they propose? Sapphorain (talk) 08:17, 9 June 2017 (UTC)

Sapphorain, after your edit the third paragraph is now nonsensical. --JBL (talk) 21:00, 10 June 2017 (UTC)

Also, this issue is not restricted to the lead. The section Arithmetic function#Neither multiplicative nor additive begins, "These important functions (which are not arithmetic functions) ...." --JBL (talk) 21:03, 10 June 2017 (UTC)

Yes, they are not arithmetical functions because they are defined on real positive numbers. I just don't understand these two last objections. If most classical introductory courses define an arithmetical function as any function whose domain are the positive integers (I cited 3 books, as well as a UIUC course in the talk page), I don't very well see how this could not be mentioned at all in the lead. Not mentioning it is equivalent to giving a wrong information. Sapphorain (talk) 19:06, 11 June 2017 (UTC)

Sorry, you are completely right, I was very confused. --JBL (talk) 23:17, 19 June 2017 (UTC)

However, I am still confused, as natural numbers (the set N of positive integers) is a subset of real positive numbers. On the other hand the precise textbook definition mentioned above, the set of arithmetical function with the operations of addition and Dirichlet convolution is a unitary ring suggests that maybe it should actually be read "inside-out" as: arithmetic functions are precisely that subset of functions in the set of N -> C functions (sequences) whose pointwise addition and Dirichlet convolution form a unitary ring. See also Dirichlet_convolution#Properties. — Preceding unsigned comment added by 82.203.239.75 (talk) 00:52, 6 January 2021 (UTC)

Arithmetic function

(Moved from my talk page:). Sapphorain (talk) 17:44, 30 July 2017 (UTC)

Dear Sapphorain, you asked for sources for the page Arithmetic function (Entropy $H_{f}$ of a given number with respect to a given multiplicative function $f$. I gave a proof, that $H_{f}$ is additive, so one does not need additional sources.

Kind regards — Preceding unsigned comment added by 88.69.187.201 (talk) 16:11, 30 July 2017 (UTC)

Dear 88.69.187.201. But yes, there definitely is a need for a reliable independent source, first about the notion you wish to introduce, and secondly about the proof you propose. Your own proof about a notion you might have invented is not an acceptable source. You appear to be making a confusion between an encyclopedia and a math blog. Kind regards. Sapphorain (talk) 17:40, 30 July 2017 (UTC)

Table

I do not understand, why Anita5912 reverted my new table. The new table is much better understandable than the old ones. It contains all functions up to x=4 and explains all the eaxmples in the text.

It also contains the prime factorization of n, which helps readers to understand the formulas. Wolfk.wk (talk) 17:05, 19 September 2018 (UTC)

I replied to your post on my talk page.—Anita5192 (talk) 17:23, 19 September 2018 (UTC)

Note odd symbol

Under the heading "First 100 values...", a hyphen is used as a multiplication symbol. — Preceding unsigned comment added by 79.77.163.188 (talk)

Indeed, bizarre. I have attempted to rectify. (There are a half-dozen other ways in which the formatting of that table is terrible; I have also replaced hyphens with minus signs for the table entries, but one could do much more.) --JBL (talk) 13:32, 29 May 2021 (UTC)

Arithmetic derivative

The notion of arithmetic (logarithmic) derivative is an old and well-known notion in number theory. See for instance (1) E. J. Barbeau, Remarks on an arithmetic derivative, Canad. Math. Bull. 4(2), 117–122 (1961); (2)V. Ufnarovski, B. Åhlander, How to differentiate a number, J. Integer Seq. 6, Article 03.3.4 (2003); (3)P. Haukkanen, J. K. Merikoski, T. Tossavainen, On arithmetic partial differential equations, J. Integer Seq. 19, Article 16.8.6 (2016). --Sapphorain (talk) 21:08, 18 May 2022 (UTC)

Too much unsourced text

Greetings Wikipedians! The sections listed below violate the Wikipedia:Verifiability policy. They contain no citations to reliable sources.

• Multiplicative and additive functions
• Notation
• {symbols} – prime power decomposition
• Some subsections in Multiplicative functions
• First 100 values of some arithmetic functions

I'll check this page in 12 months to see if the violation has been remedied. If it hasn't been fixed, I propose to delete all unsourced text. Cordially, BuzzWeiser196 (talk) 10:55, 29 September 2023 (UTC)

I disagree. The Notation section defines notation used in this article. The other sections either have adequate citations or link to other articles that define the functions.—Anita5192 (talk) 13:03, 29 September 2023 (UTC)

UserAnita5192: Thanks for clarifying. In the interest of Wiki-harmony and good fellowship, I'm willing to concede the point. Good day to you! Cordially, BuzzWeiser196 (talk) BuzzWeiser196 (talk) 12:23, 30 September 2023 (UTC)

I disagree for all sections or subsections mentioned. All are easily verifiable, and no specific source is needed. --Sapphorain (talk) 16:02, 30 September 2023 (UTC)

@Sapphorain: Greetings! When you say "all are easily verifiable", do you mean that the reader should follow links to other articles to find citations that support statements made in the Arithmetic Function article? An example would help. I am not trying to refute you. I just want to learn more about standards for verifiability. Cordially, BuzzWeiser196 (talk) 10:57, 2 October 2023 (UTC)

I mean by that that the material in the sections you list is very elementary and can be found in the various textbooks given as general references in the article: for instance Apostol’s introduction, the Hardy and Wright, the Landau, the Niven-Zuckerman-Herbert, the Bateman-Diamond. All the items in these sections are thus easily verifiable and don’t require each time a footnote citing a title and a page. But of course you are welcome to insert such footnotes if you feel like it. --Sapphorain (talk) 18:44, 2 October 2023 (UTC)

@Sapphorain: Now I understand. After reading Wikipedia:Citing sources, it seems that this article's "Further Reading" and "External Links" are what are termed "general references...that are usually found in underdeveloped articles." This article is far from underdeveloped. It's quite learned, and would benefit greatly from inline citations, which Wikipedia favors in a case like this. I wish I could help you with that task, but I don't have enough math training to take it on. My best to you! BuzzWeiser196 (talk) 19:41, 2 October 2023 (UTC)

First 100 values of some arithmetic functions

Note that in the table called "First 100 values of some arithmetic functions", there are two functions, both called lambda(n). In the code, one has a capital L and one has a small l. 2A00:23C7:9985:1701:963:1B25:CFBB:3FFA (talk) 12:46, 10 January 2024 (UTC)

Indeed; I've fixed it. Thanks for noticing! --JBL (talk) 18:32, 10 January 2024 (UTC)