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It strikes me that there is no widely accepted symbol to denote the set of usual prime numbers in $\mathbb{N}$.


$$\zeta(s)=\prod_{p\in \mathrm{?}}\frac{1}{(1-p^{-s})}$$

Wouldn't it be nicer to have a standard symbol to put in place of the "$\mathrm{?}$" instead of writing just $\Pi_p$ and specifying by words "where $p$ ranges in the set of prime numbers"?

Is there a reason for this lack of standard notation? Perhaps because primes do not form a sufficiently nice algebraic structure?

Have you seen expressive instances in the literature to define such a symbol?

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$|\operatorname{Spec}(\mathbb{Z})|$. – Moosbrugger Jul 11 '11 at 18:20
@Moosbrugger: If $|{\rm Spec}(\mathbb{Z}|$ means the set of points of the scheme ${\rm Spec}(\mathbb{Z}$, that's not so good, since ${\rm Spec}(\mathbb{Z}$ also contains the point $(0)$. OTOH, if your notation is supposed to mean the closed points, I think it's sufficiently nonstandard to confuse people. – Joe Silverman Jul 11 '11 at 18:42
I think that $|X|$ is fairly standard notation for the set of closed points of a scheme $X$. E.g., Deligne uses it in Weil I and II. – Moosbrugger Jul 11 '11 at 18:57
I think based on the general wisdom "$n$ is a number", "$p$ is a prime number" in number theory. – GH from MO Jul 11 '11 at 19:53
@Joe: $\operatorname{Maxspec}\mathbb Z$ :) – Mariano Suárez-Alvarez Jul 11 '11 at 22:02

I just write $\displaystyle \prod_{p \text{ prime}}$.

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Leaves the question: 'prime' in which structure? :) [Sorry could not resist.] – user9072 Jul 11 '11 at 19:06
@quid: well, presumably there are words before and after this and it should be clear from context. As a mild extension if I am talking about a fixed ring $R$ I think I could safely write "P a prime ideal" instead. – Qiaochu Yuan Jul 11 '11 at 19:08
I did not mean it that seriously, which I even tried to signal. In case you dislike the comment, I'll delete it (and then this one too of course). – user9072 Jul 11 '11 at 19:25
....and sometimes it is more convenient to write $\sum_{\text{even }n\ge 0} s_n$ than $\sum_{n=0}^\infty s_{2n}$. – Michael Hardy Jul 12 '11 at 19:31
@quid & @Qiaochu: Maybe there are times when you want to say $\prod_{p\text{ prime}} s_p =\text{something that depends on which structure it is}$. (But maybe I'm also taking this more seriously than I could have......) – Michael Hardy Jul 12 '11 at 19:34

To answer the actual question, I don't know any standard symbol; I've seen $P$, $\mathbb{P}$ and $\mathtt{PRIMES}$. (The last seems more common in the CS-literature, such as this famous paper.)

I would like to use this as an opportunity to make my standard plea for using multi-letter symbols; and to argue in this case for $\mathtt{PRIMES}$. There are more important concepts than can be represented by upper and lower case letters, even allowing multiple fonts and Greek letters. Moreover, multi-letter symbols are far more self-explanatory than single character ones; I can open up a paper, see $\mathtt{PRIMES}$ in the middle of a page and have a very good guess what it represents; not so with $\mathbb{P}$.

Moreover, if you tie down the simple one letter symbols for major objects, you'll won't have them available for little dummy roles like the $p$ and $s$ in your formula. For example, suppose you needed the partial product $$\prod_{\substack{ p \in \mathtt{PRIMES}\\ p < P }} \frac{1}{1-p^{-s}}$$ and needed to work with expressions like $O(\min(P^{-1} \log P, 10))$ for how things depended on your bound. (Open up pretty much any analytic number theory paper to see examples like this.) Wouldn't you be glad you hadn't wasted $P$ on a set, which is unlikely to appear in any complicated algebraic manipulations?

PS: Of course, in many cases, spelling things out in words is the best solution. There is certainly nothing wrong with "$\prod_p \ \left( \textrm{such-and-such} \right)$, where $p$ runs through the primes".

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This sounds reasonable. For the same reason, we should probably write for instance (Groups) for the category of groups instead of (Grp) or even more confusing abbreviations such as (Gp) or (Gps) (which I have all seen so far). But maybe it is a bit redundant to write such a long technical formula $p \in \mathtt{PRIMES}$ if you are not really interested in this set of all prime numbers and manipulate it by set-theoretic means. In the case of a simple product which runs over the primes, then I would prefer the notation proposed by Qiaochu (which is, of course, very popular). – Martin Brandenburg Jul 11 '11 at 20:58
Yeah, agreed. I'd have to think a little to come up with an example where using a notation for the set of primes itself was the best option. – David Speyer Jul 11 '11 at 21:17
although I find the argument to avoid "font / symbol overload" of the letter 'p' fairly compelling, I also wonder why a concept of such high importance does not yet have its own symbol. I guess, once more number theory is taught at grade-school level..... – Suvrit Jul 11 '11 at 22:43
@David: $\mathtt{PRIMES}\setminus\{2,3\}$ could be an instance. – Qfwfq Jul 12 '11 at 10:16
What about $\mathtt{PRM}$ (to not waste too much space...) ? – Qfwfq Jul 12 '11 at 10:18

I've read $\mathbb{P}$ many times and also use it.

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I've seen that one too. I've also seen $P$ and $\mathcal{P}$. I think I've even seen $\Pi$, but I don't like that one. – Pace Nielsen Jul 11 '11 at 18:20
$\prod_{p \in \Pi} f(p)$ is revolting. – Felipe Voloch Jul 11 '11 at 18:29
I think it is reasonable to reserve the \mathbb script for algebraic structures (rings, fields, and alike). Perhaps, $\mathcal P$ would be the best choice - but I don't really believe MO being influencial enough to set up a notation like this, whatever we decide here. – Seva Jul 11 '11 at 18:31
@Seva: you only have to elect me Emperor of Notation... – Mariano Suárez-Alvarez Jul 11 '11 at 22:04
@Michael: Isn't P more common for probabilities than $\mathbb{P}$? I have only a couple of books about probability; both use P. – Marius Overholt Jul 12 '11 at 6:46

As in other answers and comments: context usually suffices to explain that $p$ is a prime, whether in the rational integers or whatever. That is, when possible, no notation at all is clearer (and less bulky and visually noisy) than any possible notation.

Similarly, as I was slow to learn, objects' notations need not make explicit reference to every parameter upon which they depend: context should make most of it clear, and, if context is failing to do so, then it may be as much a complaint about the author's setting of context as anything else.

Also, as in other comments and answers, committing succinct single-letter labels for global variables is often wasteful.

Also, as computer programming teaches us, the fewer global variables the better, and, if one has such, their names should be self-explanatory, not cryptic, regardless of the illusion of "saving".

Even in situations where clarification is essential, in-lined expressions can be almost entirely prose, rather than symbolic, and displayed expressions can have a small verbal comment, as in $$ \zeta(s)\;=\;\prod_p \frac{1}{1-p^{-s}}\hskip30pt\hbox{(product over primes $p$)} $$

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At the risk of becoming 'the noise' on this question with my comments regarding 'prime': the prime elements of the rational integers are not the prime numbers. – user9072 Jul 11 '11 at 19:55
(Sometimes "noise" raises questions that need answering, because they're pervasive, even if one might argue that they "oughtn't be asked", for stylistic reason.) About "prime element" versus "primes", for example: first, saying "primes", esp. in $\mathbb Z$, by universal convention does denote the positive thingies we expect; second, context should clarify. In larger rings of integers, "primes" often comes to denote "prime ideals", so the issue changes. Or, because one needs to clarify, "prime elements mod units". And so on. Really, context should clarify, not some brittle convention. – paul garrett Jul 11 '11 at 22:24
First, in view of your parenthetical comment, let me say explictly that I certainly did not want to suggest that you were unaware of the difference. Sorry, in case it should have looked liked this. Second, I did not comment on what you responded to, but the first line. For me the by far most natural interp. of 'p is a prime, whether in the rational integers or whatever' is that p is supposed to be a prime element of the rat. integers (or whatever other structure); as 'a prime' needs to make sense in 'whatever' too. So, this seemed a bit confusing to me in a discussion on the prime numbers. – user9072 Jul 12 '11 at 0:38

For a scheme $X$, people sometimes use $\lvert X\rvert$ to denote the set of closed points of $X$. So the set of primes is $|\operatorname{Spec}(\mathbb{Z})|$ and you have:$$\zeta(s)=\prod_{p\in\lvert \operatorname{Spec}(\mathbb{Z})\rvert}\frac{1}{1-p^{-s}}.$$

This formula of course generalizes to give the $\zeta$-function of any scheme $X$ of finite type over $\mathbb{Z}$ (e.g., a variety of finite type over a finite field): $$\zeta_X(s)=\prod_{x\in\lvert X\rvert}\frac{1}{1-|\kappa(x)|^{-s}}$$ where $\kappa(x)$ is the residue field at $x$ and $\lvert\kappa(x)\rvert$ is its order.

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Minor nitpicking: $\left\vert\mathrm{Spec}(\mathbb{Z})\right\vert$ is the set of prime ideals of $\mathbb{Z}$, not the set of the actual prime numbers. – Jesko Hüttenhain Jul 11 '11 at 23:37
Also remark that usually $|X|$ denotes the underlying topological space of a scheme, whereas for varieties, where the closed points are dense, it is often meant to be the set of closed points (for example in Deligne's work on the Weil conjectures). – Martin Brandenburg Jul 12 '11 at 6:01

Since I commented so much, let me also try an answer. I find this question somehow difficult, since 'standard' seems so vague to me. [In some sense it is a very long comment, the main message is as in Martin Brandenburg's answer.]

If one asks for a true standard, then, no there is none. But, I think this is a bit much to ask for.

$\mathbb{N}$ is used in the question without explanation so a some sort of standard.

But, is it? Actually, to me there is less of a standard notation for the nonnegative and the positive integers than for the prime numbers (as cryptical expressed in a now deleted comment). Some people use $\mathbb{N}$ to mean the nonnegative integers others to denote the positve integers. (The respective other being denoted by $\mathbb{N}^{\ast}$ and $\mathbb{N}_0$ or something along these lines.) So, basically, one has to define the notation if one wishes to be sure that there is no misunderstanding; or, one uses both nonnegative and positive sufficiently frequently, because each of the pairs then makes clear which is which. [And, indeed, I spent once considerable time to figure out whether in some paper now $\mathbb{N}$ included zero or not; as I needed the result specifically for $0$ least it did.]

To wit, Henri Cohen in his recent two volume book on number theory refuses to use $\mathbb{N}$ for this reasons and uses appropriate 'decorations' of $\mathbb{Z}$ instead.

And, then there is also the question of $\mathbb{N}$ vs. $\mathbf{N}$ (and rarely, but I just saw it in the paper mentioned by David Speyer, $\mathcal{N}$). Even for the integers there is not a true standard, as the fonts vary in the same way; same for reals, complex, rationals (at least for bf vs bb).

So, why should there be a truer standard for the primes?

And if one ask for something essentially as standard as the above, then I would say yes there is: a capital P in some 'special font' blackboard bold, caligraphic, or boldface. Depending on ones choice regarding the font for, say, the integers with the exception of the concern raised by Seva or the mentioned 'notation-clashes', so that then caligraphic is not at all unusual.

Now, whether or not one should use such a symbol for the primes or not seems to me is a bit of an orthogonal discussion. I'd say it depends on the field, even on the specific paper, and the personal style. In some of my papers I do use it, in some not; depends if it seems useful.

It is true that one is much less likely to use the prime numbers as a 'building block' in some constrcut and thus needs a symbol, while say $\mathbb{Z}^n$, $\mathbb{Z}/n\mathbb{Z}$, and plenty of other things are used often.

But, then why $\mathbb{N}$ (assuming it does not contain $0$)? This could also be avoided
most of the time, or similar arguments could be made.

So, there is not a unique standard, but several standard notations (virtually all some variation on P, namely in blackboardbold, caligraphic, or boldface), which in those fields that need the notation are used frequently; to quote something recent, e.g., Iwaniec&Kowalski have $\mathbb{P}$ as a notation, just like $\mathbb{Z},\mathbb{Q},...$

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Just a tangential comment: for me $\mathbf{N}$ is poor man's $\mathbb{N}$ ! ;) – Qfwfq Jul 12 '11 at 10:24
Seriously; I think the notation $\mathbb{N}$ (or $\mathbf{N}$ for what it matters) really is the (most) standard notation for non-negative integers (I don't recall having seen it used to denote something different from $\mathbb{Z}_{\geq 0}$). For the positive integers I like the notation $\mathbb{N}^+$. – Qfwfq Jul 12 '11 at 10:31
unknowngoogle, regarding non-neg. and positive, this might be true for you and your mathematical environment. But, it is certainly not universal! Again, I can point to Iwaniec&Kowalski (and they care a lot about exposition, coventions, choice of symbols); and could point to various other places. Regarding, the bf vs bb: and, Jean-Pierre Serre says (paraphrasing) blackboard bold is for blackboards and not for print; various people agree and adhere to this. Personally I actually like bb as you, but my N does not contain 0. And each of us, is by no means alone. – user9072 Jul 12 '11 at 11:22

What about $\mathbb{N}\boldsymbol{'}$ for the set of prime numbers in the monoid $\mathbb{N}$? :)

And, of course $M\boldsymbol{'}$ for primes in a monoid $M$. It could even be shorthened to just $\boldsymbol{'}$ when it's understood that the monoid is $\mathbb{N}$.

Let's see:

$$\zeta(s)=\prod_{p\in \mathbb{N}\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$

or even just

$$\zeta(s)=\prod_{\,p \;\,\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$


$$\zeta(s)=\prod_{p \;:\,\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$

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