What does $\prod_{p}$ mean without the upper value?

Question

I was reading this paper and on page 2, I saw this: $$\prod_{p}\Bigl(\frac{1}{1-p^{-z}}\Bigr)$$ I know how to use a Pi when there is a value on the top, but what does it mean when there is no value on the top and $p$ is not assigned a value? Happy pi day!

Answer 1

Turning comments into a formal answer, the notation $\prod_p$ in the linked-to paper means the product over all primes. To be hyperprecise, it refers to the limit of the partial products when the primes are ordered by size, i.e., $\lim_{N\to\infty}\prod_{p\le N}$. This slightly cryptic notation is quite common in number-theoretic treatments of the Riemann zeta function. It's done partly for convenience -- it reduces the amount of work the typesetter has to do -- and partly for visual appeal, eliminating "clutter" from the page. (That said, some authors prefer notation such as $\prod_{p\in\mathscr{P}}$, with $\mathscr{P}$ denoting the set of primes.)

In general, any time you run into notation you don't understand, it's a good idea to look back at preceding pages to see if the notation has been used earlier; sometimes you'll find text that explains the notation. In this case, $\prod_p$ first appeared on page 2, in Lemma 2.1, in the formula

$$\zeta(z)=\prod_p{1\over1-p^{-z}}$$

There's no text that stipulates the meaning of the notation, but that is presumably because the author thought the meaning was clear from context. (You might note, they also never explicitly said the variable $p$ refers to primes.)