Talk:Arithmetic function: Difference between revisions

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== Revert ==

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I changed the article back to the previous version for the following reasons:

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* It used '''N''' for the positive integers, but we generally use that symbol for the non-negative integers, see [[natural number]].

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://1 '''Z'''⁺≡'''N'''. Non-negative are (or should be) denoted '''Z'''^*. ('''Z'''^*=0+'''N''')

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::In Wikipedia, '''N''' stands for the non-negative integers, as [[natural number]] explains. We have to use consistent conventions, even if they are arbitrary.[[user:AxelBoldt|AxelBoldt]]

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::://2 I am not quite shure if above is really so well explained in subsistent wikipedian article [[natural number]]. About '''consistent''' conventions I won't argue - according to Ribenboim (1996) this matter about what natural numbers should be is not so important and it is dependent on the usage of the definitons. Famous French "group mathematician" [[Bourbaki]] and many more used 0 as natural number like Wikipedia. Others don't. For instance [[Moebius function|Möbius function]] μ(0) is trivial and it does not show any real properties of the function itself. I am now confused even more, but I would like to use consistent stuffs in math still...

::::Yes, what counts as a "natural number", and what is denoted by '''N''', is a matter of convention. There are two competing coventions: number theorists reserve "natural number" and '''N''' for "positive integer", Bourbaki, set theorists and computer scientists reserve "natural number" and '''N''' for "non-negative integer". At one point, we in Wikipedia decided to follow the second convention. If that is not clear from [[natural number]], we have to clarify that entry. [[user:AxelBoldt|AxelBoldt]]

::::://3 Would it be OK to add somewhere in natural number a simple table like this (non-Wikipedian conventions!):

::::: ''integers'' {-∞...,-2,-1,0,1,2,...,+∞} '''Z'''

::::: ''positive integers'' {1,2,...,+∞} '''Z'''⁺ ≡ '''N'''

::::: ''non-negative integers'' {0,1,2,...,+∞} '''Z'''^* ≡ 0+'''N'''

::::: ''non-positive integers'' {-∞...,-2,-1,0} /*no designation */ ≡ '''Z'''-'''N'''

::::: ''negative integers'' {-∞...,-2,-1} '''Z'''^- ≡ '''Z'''-'''Z'''^*

:::::: No, that would not be ok, because then people could think that Wikipedia uses '''N''' for positive integers, which it doesn't. I will add symbols to [[number]]. [[user:AxelBoldt|AxelBoldt]]

* No need for colors in formulas.

://1 Red color was mentioned for unshure statement lately when cleared up put out.

::You used green for a function arrow.[[user:AxelBoldt|AxelBoldt]]

::://2 I saw two colors in formulas (blue and magenta) like this:

:::ln x ^[[e]].

:::Why other colors are so scandalous? Where can I find that I can't use colors in math formulas? We should create something like '''/mathematics/how to write formuals, etc'''. I think I am violating nothing here...

* The definition of Ψ didn't make sense: in one equation the function Ψ took two arguments, in the other it took only one. Furthermore, this does not seem to be an accepted use of the term "arithmetic function".

://1 DNA (Do not agree) Number of arguments are obviously proper property of this function. Another definition won't hurt noone. It would make him to think a little bit.

::A function either takes two arguments or one. Your definition did not make sense, because it mixed the two. It didn't specify what the ___domain and codomain of Ψ is.[[user:AxelBoldt|AxelBoldt]]

://2 Says who? A god Math Zeus a.k.a. Axel?

::I signed my statement, so you know who said it. Why play rhethorical games like this? [[user:AxelBoldt|AxelBoldt]]

::://3 I think above is not true. In the first example function &Psi;(m+n) has two parameters ''m'' and ''n'' if we watch that before adding them and it has one argument, let us say ''s''=''m''+''n''. This is OK for me. In the second term of definition I guess it was not well understood. I think (we should check this) that &Psi;(''m'',''n'') should be written as &Psi;(''m''|''n'') or as pascalike &Psi;(''m''DIV''n''). In this way function &Psi;(<font color=cyan> '''X''' </font>) has the same number of parameters, where <font color=cyan> '''X''' </font> can be whatever. About games. I didn't choose them. I advise you to check your own corrections of articles of the other authors. See bellow for the same matter on mathematical insignificance.

* "Arithmetical function" was not explained.

://1 See article.

::://2 I tried to explain that but obviously that was not explanation. Don't know what to do futher on. I saw in many ways that someone uses two different terms arithmetic(al) functions but I do not want to impose my opinions to anyone. And I have rights on my own opinions, do I?

[[user:AxelBoldt|AxelBoldt]], Thursday, April 18, 2002

://1 How to move from the NPOV in this way?

://2 It is very easy to find 1001 reasons to revert one extended (not necessarily correct) article to previous state. --[[user:XJamRastafire|XiJam]] [2002.04.18] 4 Thursday (0).

I removed this again:

''Another definition of arithmetic function is a function with exactly two [[composition|compositional properties]]:''

:&Psi;(''m''+''n'') = &Psi;[&Psi;(''m'')+&Psi;(''n'')] <tt> /composition with addition/ </tt>

:&Psi;(''m'',''n'') = &Psi;[&Psi;(''m'')&Psi;(''n'')] <tt> /composition with parameter/ </tt>

In the first equation, &Psi; is used as a function of a single argument, in the second equation it is used as a function of two arguments. This definition is mathematically meaningless. [[user:AxelBoldt|AxelBoldt]]

XJam, where did you find these two lines involving &Psi;? Maybe I can clarify it from the source. [[user:AxelBoldt|AxelBoldt]]

::::://1 At Eric's "cooking's place" http://mathworld.wolfram.com/ArithmeticFunction.html. We better go and kick his ass a little bit, ha, ha. That definition is really quite tedious, so I don't believe it would help you very much. I'll go check somewhere else.

I removed this:

:'''Note:''' Arithmetic function should not be confused with a function sometimes called ''arithmetical function'' which is in fact [[integer function]] (''f'' : '''N''' <font size=+1 color=darkgreen> &rarr; </font> '''N''').

There is no such distinction between "arithmetical" and "arithmetic" functions in the literature. Furthermore, the above arithmetical functions also qualify as arithmetic functions of course. [[user:AxelBoldt|AxelBoldt]]

::://1 No comment. --[[user:XJamRastafire|XJam&Psi;]] [2001.04.19] 5 Friday (0).

:::::AO --[[user:XJamRastafire|XJam&tau;]] [2001.04.19] 5 Friday (1st ed).

== Arithmetic vs arithmetical ==

please, if concensus is reached could this be clarified in the article ?

*The page's title is "arithmetic f." (and that where most pages link to).

*The first paragraph (the only which is displayed with [navigation popup]s e.g.) only defines "number theoretic f.",

*The second paragraph defines "arithmetical f."

*The rest of the page only speaks of "arithmetic f."

&mdash; [[User:MFH|MFH]]:[[User talk:MFH|Talk]] 17:43, 2 April 2007 (UTC)

== Changes ==

I've made some changes to the article. The introduction was not general enough. I made it general and simple. The old introduction made only passing reference to more complicated objects such as Dirichlet convalutions. For that reason I removed a lot of it. It was difficult to add more detail later without repeating oneself. That problem's gone now. Please feel free to add new sections. [[User:Declan Davis#top|<span style="background-color:green;color:gold;">&nbsp;<i><b>Declan Davis</b></i>&nbsp;</span>]] [[User talk:Declan Davis#top|<span style="background-color:green;color:gold;">&nbsp;<i><b>(talk)</b></i>&nbsp;</span>]] 20:58, 1 October 2008 (UTC)

== Examples of arithmetic functions ==

I've reinstated "of arithmetic functions" in the section heading. Since it follows directly after the definitions of multiplicative and completly multiplicative functions I want to make it clear that they are not example of these two. [[User:Declan Davis#top|<span style="background-color:green;color:gold;">&nbsp;<i><b>Δεκλαν Δαφισ</b></i>&nbsp;</span>]] [[User talk:Declan Davis#top|<span style="background-color:green;color:gold;">&nbsp;<i><b>(talk)</b></i>&nbsp;</span>]] 11:46, 3 October 2008 (UTC)

== Parsing error? ==

In the divisor convolution subsection, I see a parsing error for the sigma_5 and sigma_7 formulas when viewing the whole page, but not when in preview of just that section. The same thing is happening in the class number section. I cannot figure out what is going on. [[User:JoshuaZ|JoshuaZ]] ([[User talk:JoshuaZ|talk]]) 14:06, 14 February 2014 (UTC)

:Now, there would be no persing error because I have recompiled this article. Perhaps it might be temporal errors. --[[User:Enyokoyama|Enyokoyama]] ([[User talk:Enyokoyama|talk]]) 15:08, 14 February 2014 (UTC)

== 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.--[[User:GangofOne|GangofOne]] ([[User talk:GangofOne|talk]]) 03:57, 24 May 2014 (UTC)

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== Confused ==

In [[Arithmetic function#Miscellaneous]], in the third to fifth lines, on my computer, the passage is very confused. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/38.117.79.31|38.117.79.31]] ([[User talk:38.117.79.31|talk]]) 11:28, 10 July 2014 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

: Yes, I modified a bit this strange layout. But the layout of the whole section is still very unconventional. [[User:Sapphorain|Sapphorain]] ([[User talk: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.

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== 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? <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Luqui|Luqui]] ([[User talk:Luqui#top|talk]] • [[Special:Contributions/Luqui|contribs]]) 23:50, 7 June 2017 (UTC)</small> <!--Autosigned by SineBot-->

: 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:

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:: ... I added three classical references giving this less restrictive definition, and modified the lead accordingly. [[User:Sapphorain|Sapphorain]] ([[User talk: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). {{u|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)? --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|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). [[User:Sapphorain|Sapphorain]] ([[User talk: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.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 22:04, 8 June 2017 (UTC)

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

:::{{u|Sapphorain}}, after your edit the third paragraph is now nonsensical. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|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) ...." --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|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. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 19:06, 11 June 2017 (UTC)

::::: Sorry, you are completely right, I was very confused. --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|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]]. <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/82.203.239.75|82.203.239.75]] ([[User talk:82.203.239.75#top|talk]]) 00:52, 6 January 2021 (UTC)</small> <!--Autosigned by SineBot-->

== Arithmetic function ==

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

Dear Sapphorain, you asked for sources for the page Arithmetic function (Entropy <math>H_f</math> of a given number with respect to a given multiplicative function <math>f</math>. I gave a proof, that <math>H_f</math> is additive, so one does not need additional sources.

Kind regards <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/88.69.187.201|88.69.187.201]] ([[User talk:88.69.187.201#top|talk]]) 16:11, 30 July 2017 (UTC)</small> <!--Autosigned by SineBot-->

: Dear [[Special:Contributions/88.69.187.201|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. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain#top|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. [[User:Wolfk.wk|Wolfk.wk]] ([[User talk:Wolfk.wk|talk]]) 17:05, 19 September 2018 (UTC)

:I replied to your post on my talk page.—[[User:Anita5192|Anita5192]] ([[User talk: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. <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/79.77.163.188|79.77.163.188]] ([[User talk:79.77.163.188#top|talk]]) </small>

: 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.) --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|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). --[[User:Sapphorain|Sapphorain]] ([[User talk: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, [[User:BuzzWeiser196|BuzzWeiser196]] ([[User talk: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.—[[User:Anita5192|Anita5192]] ([[User talk: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, [[User:BuzzWeiser196|BuzzWeiser196]] ([[User talk:BuzzWeiser196|talk]]) [[User:BuzzWeiser196|BuzzWeiser196]] ([[User 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. --[[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 16:02, 30 September 2023 (UTC)

::{{ping|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|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, [[User:BuzzWeiser196|BuzzWeiser196]] ([[User talk: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. --[[User:Sapphorain|Sapphorain]] ([[User talk: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! [[User:BuzzWeiser196|BuzzWeiser196]] ([[User talk: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. [[Special:Contributions/2A00:23C7:9985:1701:963:1B25:CFBB:3FFA|2A00:23C7:9985:1701:963:1B25:CFBB:3FFA]] ([[User talk:2A00:23C7:9985:1701:963:1B25:CFBB:3FFA|talk]]) 12:46, 10 January 2024 (UTC)

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