Talk:Arithmetic function

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 Field: Number theory


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

  • It used N for the positive integers, but we generally use that symbol for the non-negative integers, see natural number.
//1 Z+N. Non-negative are (or should be) denoted Z*. (Z*=0+N)
In Wikipedia, N stands for the non-negative integers, as natural number explains. We have to use consistent conventions, even if they are arbitrary.AxelBoldt
//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 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. 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. 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.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.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? AxelBoldt
//3 I think above is not true. In the first example function Ψ(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 Ψ(m,n) should be written as Ψ(m|n) or as pascalike Ψ(mDIVn). In this way function Ψ( X ) has the same number of parameters, where X 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?

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. --XiJam [2002.04.18] 4 Thursday (0).

I removed this again:

Another definition of arithmetic function is a function with exactly two compositional properties:

Ψ(m+n) = Ψ[Ψ(m)+Ψ(n)] /composition with addition/
Ψ(m,n) = Ψ[Ψ(m)Ψ(n)] /composition with parameter/

In the first equation, Ψ 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. AxelBoldt

XJam, where did you find these two lines involving Ψ? Maybe I can clarify it from the source. AxelBoldt

//1 At Eric's "cooking's place" 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 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. AxelBoldt

//1 No comment. --XJamΨ [2001.04.19] 5 Friday (0).
AO --XJamτ [2001.04.19] 5 Friday (1st ed).

arithmetic vs arithmetical[edit]

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."

MFH:Talk 17:43, 2 April 2007 (UTC)


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.  Declan Davis   (talk)  20:58, 1 October 2008 (UTC)

Examples of arithmetic functions[edit]

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.  Δεκλαν Δαφισ   (talk)  11:46, 3 October 2008 (UTC)

Parsing error?[edit]

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. 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. --Enyokoyama (talk) 15:08, 14 February 2014 (UTC)

error in formula[edit]

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)


In Arithmetic function#Miscellaneous, in the third to fifth lines, on my computer, the passage is very confused. — Preceding unsigned comment added by (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)