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Counting primes of the form $S_1(a_n)$ vs primes of the form $S_2(b_n)$

Let $n$ be an integer $>1$. Let $S_1(a_n)$ be a symmetric irreducible integer polynomial in the variables $a_1,a_2,...a_n$. Let $S_2(b_n)$ be a symmetric irreducible integer polynomial in the ...
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Is the application of $\mu$ on $P_x(s)^k$ analogous to the differentiation $\frac{d^k f(\lambda) }{d\lambda^k}\biggr|_{\lambda=0}$?

Let me start with the following on elementary symmetric polynomials: The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity ...