User talk:Peter Luschny/EulerTotient

Compare Euler's totient function

$\varphi (n)=n\cdot {\frac {\prod _{p\mid n}(p-1)}{\prod _{p\mid n}p}}=n\cdot {\prod _{p\mid n}{\bigg (}1-{\frac {1}{p}}{\bigg )}}.\,$

with the Dedekind psi function

$\psi (n)=n\cdot {\frac {\prod _{p\mid n}(p+1)}{\prod _{p\mid n}p}}=n\cdot {\prod _{p\mid n}{\bigg (}1+{\frac {1}{p}}{\bigg )}}.\,$

— Daniel Forgues 01:53, 29 November 2010 (UTC)

One may thus define this arithmetic function (not usually giving integers)

${\frac {\psi (n)}{\varphi (n)}}=\prod _{p\mid n}{\frac {p+1}{p-1}}\,$

Is this a function of interest? — Daniel Forgues 23:59, 7 August 2011 (UTC)

User page versus wiki page

Dear Daniel,

I do understand "The OEIS Contributor's License Agreement" and I do understand that it may be edited, altered, or removed by other contributors.

This said I think it is a good idea to make a difference between a user page and an official wiki page. My policy is to respect a user page as a place for the user to express the way he thinks it is adequate to present a subject. Here a much more subjective style is acceptable than on an official wiki page. If you think there is something worth on this page to be presented in a more general and objective setup go ahead and copy it to the page of the official wiki.

You will notice the difference between my style on my user pages (which I sometimes call my 'Blog on OEIS') and on pages which I contribute to the wiki like 'Eulerian polynomials'. There I try to write in a style more suited for a collaboratively written encyclopedia, excluding my personal opinions and beliefs.

Regarding your edit: There is a page on the wiki "Euler's totient function" as you know since you are the main contributor to this page. And the formula you added here is there, so I see no real need to add it here. But for me it is misplaced here because it was not my intention to show all the (in this case trivial) reformulations of this formula; my intention was to focus on a special way to split the formula which is reflected by sequences in the database. To keep this focus I will delete your addition in a future edit of this page. Peter Luschny 14:12, 29 November 2010 (UTC)

I apologize. I admit and understand that I should not do edits, however small, on user pages. User talk pages are the proper location to share ideas with users. Thanks for your advice. — Daniel Forgues 02:59, 30 November 2010 (UTC)

Typo in formulae

In the following, the bounds should be $\scriptstyle 1\,\leq \,k\,\leq \,n\,$

$\varphi (n)=\sum _{1<k<n}\left[k\perp n\right].$

Since a and b have only one divisor in common this is also the greatest common divisor, (a,b) = gcd(a,b) = 1.

$\varphi (n)=\sum _{1<k<n}\left[{\text{gcd}}(k,n)=1\right].$

— Daniel Forgues 00:07, 8 August 2011 (UTC)

Ups, thanks for observing this. I think I wanted to write

$\varphi (n)=\sum _{0<k<n}\left[k\perp n\right].$

But there is also this confusing $\scriptstyle \left[n\perp n\right].$

— Peter Luschny 20:49, 22 August 2011 (UTC)