# About an equation for the partial Euler product in terms of elementary symmetric polynomials

With Elementary Symmetric Polynomials we can construct the following equation:

$$\frac{1}{(p_n\#)^s}\sum_{k=0}^n (-1)^{k+n} e_k(2^s,3^s,5^s,7^s,\dots,{p_n}^s)=\prod_{k=1}^n\left(1-\frac{1}{{p_k}^s}\right)$$

which equals $$1/\zeta(s)$$ as $$n\to\infty$$.

What does this formula imply about the Riemann Zeta function? Are there any papers that reference or make use of this formula so I could learn about it?

• First thing, this is only converges for some values of $s$. It tells us nothing new about $\zeta(s)$ – Somos Oct 17 at 22:25