Talk:Coprimorial

One possible notation might be $\phi !(n)$, though perhaps that could be confused for $\Gamma (\phi )n$ or $(\phi (n))!$. Alonso del Arte 20:43, 28 November 2010 (UTC)

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(1) It seems to me more clear if we use one definition (and only one definition) in one line and an explanation of different notations in another line if this is necessary. In particular this avoids the impression that mathematical identities are displayed.

(2) It does not make sens to repeat over and over again trivial notational variants especially in a wiki which is so heavily cross-referenced as this one. For instance it is sufficient to explain the equivalence of the notational variants $\scriptstyle k\perp n\,$, gcd(k,n)=1 and (k,n)=1 once, say at Coprimality.

(3) I think the use of the equiv symbol should be reserved to the use in modular arithmetic which is so important for many notions used by number theory; i.e. to the congruence relation.

(4) The term "orthogonal numbers" as used here is not good. Let me ask: Is 3 an orthogonal number? Orthogonality is a relation, not an attribute! Knuth, Graham and Patashnik call it "k is prime to n".

Here my proposal which avoids much of the clutter:

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

Here $\scriptstyle [\cdot ]\,$ is Iverson bracket, $\scriptstyle k\perp n\,$ means k is prime to n (see coprimality) and $\varphi \,$ reminds of Euler's totient function

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

Since

$\varphi (n)=\sum _{d\mid n}d\ \mu \left({\frac {n}{d}}\right)$

where µ is the Möbius function we also have

$\varphi _{_{_{!}}}(n)={n^{\varphi (n)}}\prod _{d|n}{\bigg (}{\frac {d!}{d^{d}}}{\bigg )}^{\mu {\big (}{\tfrac {n}{d}}{\big )}}.$

The trivial fact

$\prod _{1\leq k\leq n}k=\prod _{1\leq k\leq n}k^{\left[k\perp n\right]}\prod _{1\leq k\leq n}k^{\left[k\not \perp n\right]}$

can also be written as $\scriptstyle n!\ =\ \varphi _{_{_{!}}}(n)~{\overline {\varphi }}_{_{_{!}}}(n)\,$ where $\scriptstyle {\overline {\varphi }}_{_{_{!}}}(n)$ is the noncoprimorial of n.

Peter Luschny 13:45, 22 July 2011 (UTC)

Good ideas all. I'd execute them myself, but you can do it, too, Peter. Alonso del Arte 16:51, 22 July 2011 (UTC)

I've already made some of those changes. I will also use the Iverson bracket instead of the Kronecker delta throughout the wiki, as is recommended in User:Peter Luschny/NotationMatters. I've already begun to do so. For definitions, the right notation would be to use $\scriptstyle :=\,$ instead of $\scriptstyle \equiv \,$, but the former is less visually appealing (although more precise.) Mathworld uses $\scriptstyle \equiv \,$. Should we use $\scriptstyle :=\,$ throughout. Also, I think I prefer $\scriptstyle k\,=\,1\,$ to $\scriptstyle n\,$, as in

$\prod _{k=1}^{n}k={\Bigg (}\prod _{k=1}^{n}k^{\left[k\perp n\right]}{\Bigg )}~{\Bigg (}\prod _{k=1}^{n}k^{\left[k\not \perp n\right]}{\Bigg )}\,$

to

$\prod _{1\leq k\leq n}k=\prod _{1\leq k\leq n}k^{\left[k\perp n\right]}\prod _{1\leq k\leq n}k^{\left[k\not \perp n\right]}.\,$

Which is best?

— Daniel Forgues 21:50, 22 July 2011 (UTC)

I too prefer {k = 1}^n, it says to me that you start your iterator at a low value and rise it to the ending value. Alonso del Arte 23:41, 22 July 2011 (UTC)

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With regard to indices: To me mixing of {k = 1}^n with a stack notation as in

$\varphi _{_{_{!}}}(n)=\prod _{\stackrel {i=1}{i\perp n}}^{n}i,$

is a little confusing. In such a case I do prefer to see the constraints in a single place.

$\varphi _{_{_{!}}}(n)=\prod _{\stackrel {1\leq i\leq n}{i\perp n}}i,$

The use of := to emphasise a definition is very much a matter of taste and as Daniel said is visually not very appealing. Therefore most authors do not use it at all but just make it clear from the context. In fact not a single mathematical book on my desk uses the ':=' notation. If one wishes to emphasize a definition it is more important to give it a line for itself and not mix/chain it with other expressions; if this is not enough emphasize one can also use something like

Def: a = b.

But let's be open to variations ;-) — Peter Luschny 07:24, 23 July 2011 (UTC)

What I like in separating the range (of summation, product, ...) from the other constraints, as in

$\prod _{k=1}^{n}k={\Bigg (}\prod _{k=1}^{n}k^{\left[k\perp n\right]}{\Bigg )}~{\Bigg (}\prod _{k=1}^{n}k^{\left[k\not \perp n\right]}{\Bigg )}\,$

is that

$k^{\left[k\perp n\right]}\,$

and

$k^{\left[k\not \perp n\right]}\,$

appear as functions. The constraints expressed via the Iverson bracket could even be used in a definite integral, e.g.

$\int _{0}^{n}x[x\in \mathbb {Q} ]dx=0,\quad n\in \mathbb {R} ,\,$

making for a uniform notation. What do you think? — Daniel Forgues 19:28, 24 July 2011 (UTC)

If I may have a George Will moment here: in music, most younger performers have gotten used to the idea that if a composer does not provide a dynamic marking for a passage after a long rest, then the same dynamic of the previous passage remains in force. After all, the younger performers understand that in the notation software's computer playback which the composer may have relied on (but shouldn't), this is the case.

In all fields, computer notions will have an increasing effect on how humans parse things. Alonso del Arte 19:44, 24 July 2011 (UTC)