Talk:Euler's totient function

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 A nand Sadasivamt@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

Consistency of Notation
I think it is imperative that this article choose a consistent notation for phi. This has been mentioned before on the talk page (Different Phis and Letter Phi in Mathematics), however, it does not appear to be completely resolved. It is completely reasonable for a sentence in the intro to note, as it does now, that the totient "is written using the Greek letter phi as φ(n) or ϕ(n)". However, throughout the article, using different typeset versions of phi causes needless confusion for readers. I'd recommend the latex typeset capital phi (ϕ) since this appears to the be the standard in a wide variety of current works (e.g., , pg. 8). TripleShortOfACycle (talk - contribs) - (she/her/hers) 07:32, 7 April 2021 (UTC)
 * The article has used φ for a long time. Please leave it like it is, changing notation is pointless and bound to produce heated and sterile arguments. There were 4 uses of ϕ instead of φ, I changed them for consistency, as φ is overwhelmingly more common in the article. Note that both are lowercase phi, the uppercase version is Ф instead. Tercer (talk) 08:30, 7 April 2021 (UTC)
 * Thanks for making the change! I am more concerned with consistency than which version was ultimately chosen. TripleShortOfACycle (talk - contribs) - (she/her/hers) 11:53, 7 April 2021 (UTC)

Perhaps your browser is rendering the characters differently than mine? For me $φ(n)$ gives me the loopy character φ(n), or in math mode $$\varphi(n)$$, and $ϕ(n)$ gives me the straight character ϕ(n), or in math mode $$\phi(n)$$. Do you get something different? In that case it's fine to change to math mode, so that all readers can see the difference, but then please also do it for $φ(n)$, otherwise it looks ugly. Tercer (talk) 10:52, 7 April 2021 (UTC)
 * Yes, I will change to math mode: I don't get at all what you describe (I am on MacOS Big Sur)! Thank you! --Sapphorain (talk) 11:25, 7 April 2021 (UTC)
 * The two variants of phi look correct in math mode on my browser: $φ, ϕ$. However, the first Unicode variant appears loopy on my edit window but is rendered as straight (with a shorter vertical bar) if not included into math: φ, ϕ. A further reason for avoiding raw html for mathematics. D.Lazard (talk) 11:37, 7 April 2021 (UTC)
 * I'm on Firefox on Ubuntu, it renders φ, ϕ correctly in all cases. Doesn't matter, we need it to render correctly for all readers, and apparently only does the job. Tercer (talk) 11:45, 7 April 2021 (UTC)

Proof for the multiplicity of the totient function
I think it would be better is a full proof for the multiplicity of the totient function is furnished in this page. If someone disagrees, please say so. I will put up the proof 3 days after this message is posted if there is no opposition — Preceding unsigned comment added by NKRVVI (talk • contribs) 10:49, 16 September 2021 (UTC)
 * What is the multiplicity of the totient function? D.Lazard (talk) 10:58, 16 September 2021 (UTC)
 * If you mean multiplicative property of the totient function, the proof sketched in seems sufficient. D.Lazard (talk) 11:02, 16 September 2021 (UTC)

There is a proof outline. I think that with a little more explanation, it could become a full proof.NKRVVI (talk) 03:22, 17 September 2021 (UTC)
 * We generally only include proof outlines, and not full proofs, for most articles. Proofs are typically included only when they are particularly enlightening, and not merely as a way to verify a mathematical truth. In most cases, an outline suffices for that. For verification, we can point to mathematical publications that contain the proofs. —David Eppstein (talk) 05:41, 17 September 2021 (UTC)

new reference
There is a new reference (currently #25) to StackExchange. StackExchange is not considered a reliable source. Bubba73 You talkin' to me? 07:46, 23 November 2021 (UTC)
 * The other reference for the same claim, "Bordellès in the external links", refers to a long-removed deadlink, archived at https://web.archive.org/web/20120301060141/http://www.les-maths.net/phorum/read.php?5,359275,359275 but possibly not in the right version, which as an open forum also looks non-reliable. —David Eppstein (talk) 08:02, 23 November 2021 (UTC)