Product Notation Conventions

Question

If a product has the following notation, does that mean the product has zero terms? If there is an expression after this symbol, is the entire quantity, with the product symbol included, equal to zero? What is the proper notation for this, if the product must always start at g=1? $$\prod_{g=1}^{0}$$

Answer 1

The empty product is $1$. Since $\prod_{a\in S}a=\frac{\prod_{c\in S\cup T}c}{\prod_{b\in T}}$ for disjoint sets $S,\,T$, the choice $S=\emptyset,\,0\not\in T$ gives $$x:=\prod_{b\in T}\ne 0\implies\prod_{a\in\emptyset}a=\frac{x}{x}=1.$$This, for example, is the logic behind $0!=1$, or the "product" $\prod_{a\in\{d\}}a=d$ of one object $d$.