A symbol to denote the set of prime numbers ? It strikes me that there is no widely accepted symbol to denote the set of usual prime numbers in $\mathbb{N}$.
Look: 
$$\zeta(s)=\prod_{p\in \mathrm{?}}\frac{1}{(1-p^{-s})}$$
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?

 A: 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.
A: 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 $0$...at 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 $\mathbb{Z},\mathbb{Q},...$
A: I just write $\displaystyle \prod_{p \text{ prime}}$. 
A: I've read $\mathbb{P}$ many times and also use it.
A: 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".
A: 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 
$$
\zeta(s)\;=\;\prod_p \frac{1}{1-p^{-s}}\hskip30pt\hbox{(product over primes $p$)}
$$
A: 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
$$\zeta(s)=\prod_{\,p \;\,\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$
or 
$$\zeta(s)=\prod_{p \;:\,\boldsymbol{'}}\frac{1}{(1-p^{-s})}$$
A: The set of prime numbers, though not having fixed or regular structure within it, should be symbolized as N' either in Boldface or Blackboard Bold type with the apostrophe or prime symbol to its right as a superscript denoting that set or subset of natural numbers (N) as being that of PRIME numbers.
