Talk:Arithmetic function: Difference between revisions

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== 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."

— [[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 ==

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:::: 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 ==

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: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 vs arithmeticalderivative ==

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)

== ExamplesFirst 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)