Talk:Euler's totient function

The name of the function

I am an American number theorist, and I cannot find a single book in my shelves that calls the Euler phi function the "totient" or "Euler totient" function. It is in all cases referred to as the Euler phi function. I think for a function as well-used as this one, especially by undergraduates, we should use standard terminology. It is hard to cite all the references -- please look at any number theory book on your shelf. I am curious about what other readers think.Jaatex (talk) 19:19, 11 April 2015 (UTC)

I am an American number theorist, and in no reference on my shelf is the Euler phi function called the totient function or the Euler totient function. With a function as well-used as this, especially by undergraduates, shouldn't we use the standard name? I'm curious how others feel. Jaatex (talk) 19:27, 11 April 2015 (UTC)

I am also a number theorist, and have very often heard the phi function called the "totient function", or "Euler totient function". The term was apparently invented by Sylvester. It is mentioned by Niven and Zuckerman (3rd edition, p.22). In the "Handbook of estimates in the theory of numbers" (B. Spearman, K.S. Williams, Carleton Mathematical Lecture Note 14, 1975), on the definition p. 4, phi is called "Euler's totient function". If you search on MathSciNet, you will find that the number of articles containing in their titles "Euler totient function" is 34 (73 for "Euler phi function"), and that the number of referee's reports citing "Euler totient function" is 386 (225 for "Euler phi function"). (Searching for just "totient function" is misleading: there are many other totient functions). Sapphorain (talk) 20:57, 11 April 2015 (UTC)

Thanks for the research. As you found, two-to-one in titles it is referred to as the Euler phi function. My bookshelf research: Ireland and Rosen, "A Classical Introduction to Modern Number Theory" and Pierre Samuel (translated from French), "Algebraic Theory of Numbers"; Kenneth H. Rosen "Elementary Number Theory"; Joseph H. Silverman, "Number Theory"; David M. Burton, "Elementary Number Theory" (the only reference where "totient" is mentioned: "The function \phi is usually called the \it {Euler phi-function} (sometimes, the \it indicator or \it totient)"); Serge Lang, "Algebra"; and "Andre Weil, "Number Theory, An approach through history...". In the above references, totient is neither in the index nor the text (except as noted in Burton, where it is not in the index). I understand that totient function terminology is sometimes used, and that it is historical. My point is that it is not commonly used. If a student tries to look up the totient function in a standard text, they will find nothing. I'm not sure why totient was chosen for the title of this article. I would like to change it to the more common name. Jaatex (talk) 14:39, 15 April 2015 (UTC)

In the past, this page was named "Euler's phi function". It is not clear when the name has changed, but the first line has been changed in 2003, apparently without any discussion, by this edit [1]. In any case, as both Phi function and Euler's phi function redirect to this article, the reader searching for this article will find it easily. Personally, I have no opinion on the best title nor for the name that should be used in the article (as far as both names appear in the first sentence). D.Lazard (talk) 15:23, 15 April 2015 (UTC)

Thank you for this information. This is my first attempt to edit a Wikipedia page. I see the edit from 2003, thanks for pointing it out. I do not see why the change was made. How was the change made without discussion? How does one go about getting it changed back? Do people vote?  :-) Jaatex (talk) 15:54, 15 April 2015 (UTC)

In Wikipedia, decisions are taken by consensus, as described in details in WP:CON. However, few people have given their opinion. One could start a WP:request for comments. But, as nobody disagrees formally with you, and your arguments are convincing, I have self-reverted my revert of your edits. The change of title (called move in WP) seems less important than editing the content, because the title you suggest exists as a WP:redirect. For technical reasons, it may be done (in this case) only through a WP:move request. D.Lazard (talk) 19:12, 15 April 2015 (UTC)

Totient numbers

This section contains there are infinitely many nontotients, and indeed every odd number has an even multiple which is a nontotient, the word "even" being added by a recent edit. In both versions, this sentence is a nonsense, as every odd number greater than 1 is a nontotient; thus no need to consider multiples. I guess that the correct assertion should be every totient has a multiple (by an odd number) that is a nontotient. However this needs to be checked on the source. D.Lazard (talk) 09:59, 25 April 2015 (UTC)

As you say, odd numbers greater than 1 are trivially nontotients, so what is of interest is the existence of even nontotients. I imagine the previous writer meant "there are infinitely many even nontotients, and indeed..." but just forgot the "even". I've checked the paper and in fact it proves that any number (even or odd) has a multiple which is a nontotient. If n is odd then the nontotient multiple of 2n gives an even nontotient multiple of n, so this is equivalent to saying any number has an even nontotient multiple. I'll make those changes. Especially Lime (talk) 08:54, 20 May 2016 (UTC)

Unexplained revert of style improvement

My edit in Euler's totient function#Euler's product formula has been reverted by an IP user, without any explanation. My edit consisted in

• Removing, per MOS:HEADINGS the redundant reference (through a formula) to the article title.
• Replacing a heading consisting of a technical formula by a less technical phrase (the formula was redundant, as reproduced in the body
• Avoiding the confusing term "modulo-and-coprime", which is nowhere defined in Wikipedia
• Linking coprime

All are style improvements that does affect in any way the content of the article. As I cannot find any valid reason for rejecting these edits, I have restored them. Please, if I have missed something, please discuss here before a second revert. D.Lazard (talk) 15:52, 13 January 2017 (UTC)

Not Empty Set is that Correct - Puzzling

Empty set says (and not related in any way to the Greek letter Φ), but inspired by the letter Ø in the Norwegian and Danish alphabets.

My last month Mathematics Today Magazine from mtG has a sentence,-

"Empty set is subset of every set and every set is subset of itself. We denote by it by Φ or {}"

is that phi is small case, couldn't categorize on that font whether it is lower or upper alphabet.

And however, Φ(n) -> Phi(n) -> Euler's totient function

Maybe we can see through Article, and make reference of Empty set if it is relevance to do that as per the sentence made by mtG

—Dev Anand Sadasivam^t@lk 02:26, 30 June 2018 (UTC)

Letter Phi in mathematics

(From my talk page, + answer, Sapphorain (talk) 22:23, 26 December 2018 (UTC))

Hello, regarding your recent revert of my edit, could you provide me an example of some respectable mathematical writing where the author would use the glyphs "φ" and "ϕ" as different variables (to denote two different things)? --Alexey Muranov (talk) 22:08, 26 December 2018 (UTC)

No I cannot, and I am not interested in finding one. But it is not the point. Two different ways of writing the same letter have been used in many instances to denote different objects. So the present precision is quite legitimate and there is no reason to suppress it. Sapphorain (talk) 22:23, 26 December 2018 (UTC)

I agree with showing the two ways it is commonly written. Bubba73 ^{You talkin' to me?} 23:25, 26 December 2018 (UTC)

Although they both refer to the Greek letter phi, the two are separate symbols. I believe they both should be shown or described. In fact, I have seen some fields of mathematics prefer one symbol to the other.—Anita5192 (talk) 23:28, 26 December 2018 (UTC)

Divisor Sum Proof Error

A line in the "Divisor Sum" portion under "Computing Euler's totient function" states "Any such k must clearly be a multiple of n/d, but it must also be coprime to d," but there are many fractions in the n = 20 example given where k is not coprime to d. For example, k = 2 and d = 10 are not coprime. I didn't know how to flag content for review, so I posted on the talk page. X9du (talk) 21:39, 3 May 2020 (UTC)X9du