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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 $ [f(\lambda)=]\prod_{j=1}^n ( \lambda-X_j)=\lambda^n-e_1(X_1,\ldots,X_n)\lambda^{n-1}+e_2(X_1,\ldots,X_n)\lambda^{n-2}-\cdots+(-1)^n e_n(X_1,\ldots,X_n). $

Let $n$ be even. The $k$th derivative evaluated at $\lambda=0$ is given by $ \frac{d^k f(\lambda) }{d\lambda^k}\Biggr|_{\lambda=0}=(-1)^ke_{n-k}(X_1,\ldots,X_n) $


So far. Now to something completely different: Take the $k$th power of the truncated Prime Zeta function $ P_x(s)^k=\left(\sum_{p_t\,\in\mathrm{\,primes}\leq x} p_t^{-s}\right)^k =\sum_{k_1+k_2+\cdots+k_m=k} {k \choose k_1, k_2, \ldots, k_m} \prod_{1\le t\le m}p_{t}^{k_{t}}\;, $ where ${k \choose k_1, k_2, \ldots, k_m} = \frac{k!}{k_1!\, k_2! \cdots k_m!} $, according to the Multinomial theorem, e.g. by W|A.

Let me show you the relevant part of the example $ (11^s+13^s+17^s+19^s+23^s)^4=\ldots+24\cdot( 46189^s+55913^s+62491^s+ 81719^s+96577^s)+\ldots $ where $46198=11\cdot 13\cdot 17\cdot 19$ and so on, in fact this part is the elementary symmetric polynomial $e_4(X_1,\ldots,X_5)$ with $X_k=p_{k+4}$ (the offset is just to not mix up the multinomial coefficient with the almost-primes).


Ok, now one more thing: The wiki page on Incidence Algebras states:

Multiplying by μ is analogous to differentiation.



Fine, now if I could multiply or apply (where I neglect the multinomial coefficients for the moment) the standard Möbius function $\mathbf \mu$ by the $k\,$th power of the truncated Prime Zeta function the only terms that survive are those, where all prime powers shows up only once, like in $46189^s$. All others have at least a square or a higher power of at least one prime and are therefore nulled out. Or more general, given a series $p_n(s)=\sum_{k=1}^n a_k $. I'd like to get $\hat{p}_n(s)=\sum_{k=1}^n \mu(k)a_k $. We are left with the elementary symmetric polynomial mentioned before.

So here is my question (again):

Is the application of $\mu$ on $P_x(s)^k$ analogous to the differentiation $\frac{d^k f(\lambda) }{d\lambda^k}\biggr|_{\lambda=0}$ ?

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    @joriki you are right. I've gotta think about where to put this to be more visible and clear. Anyway, anyidea?2012-10-30

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