I show below a formula that I've derived recently from the well-known Euler product formula, which could be considered as a generalization of it.
Let's start with a definition. For any non-empty set of primes $A \subseteq \mathbb{P}$, I define $A^{\otimes}$ as the set of numbers smooth over $A$, that are the naturals having all their prime divisors in $A$. For the ease of my development, I'll consider arbitrarily that $1 \in A^{\otimes}$.
Here is my proposition: for any non-empty set $A \subseteq \mathbb{P}$, for any real $a$ such that $ { 0 < a < \min(A) } $ and for any real $s > 1$, $$ \prod _{p \in A} \frac{1}{1-(a/p)^s} \; = \; \sum_{n \in A^{\otimes}} \left( \frac{a^{\Omega(n)}}{n}\right)^s $$ where $\min(A)$ is the smallest element of $A$ and $\Omega (n)$ is the total number of prime factors of $n$ (following usual definitions).
I haven't made a formal proof so far but this formula appears not too difficult to derive if you follow the original Euler's reasoning—I can provide details, if needed. Now, my claim of "generalization" should appear clear: with $A \triangleq \mathbb{P}$ and $a \triangleq 1$, noticing that $\mathbb{P}^{\otimes} = \mathbb{N}^+$, the formula above boils down to the exact Euler product formula.
My questions:
- Does the formula above sound valid (e.g. missing assumptions, ...)?
- Is this formula, or possibly a variant of it, a known/published result?
- Is there any "usefulness" in such formula, e.g. in the study of Riemann's $\zeta$ function?
PS: To be complete about point 2., I mention that a less general formula (where $a \triangleq 1$) appears to be known at least, simply because it has been discussed in the MO question Series of reciprocals of smooth numbers. However, I'd appreciate a reference (book/paper) for this result, if any.
