What is known about a kind of dual version of the Riemann zeta function defined $\bar{\zeta}(n)$ as follows:
If there is a duality between the primes and non-trivial Riemann zeta-zeroes. We can convert $$ \zeta(s) = \prod_{p} \frac{1}{1-p^{-s}} $$ for prime $p$, into a new function $$ \bar{\zeta}(n)=\prod_{\rho}\frac{1}{1-n^{-\rho}} $$ where the product is now over non-trivial zeroes. By plotting this for a finite number of zeroes it appears that the new function goes to zero at the primes, i.e. the primes are the new zeros.
There is a note describing this here.
Question 2: Is there a consistent duality of the natural numbers. I.e. simplistically speaking if $$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} $$ can we find an expression $$ \bar{\zeta}(n)= \sum_{\gamma} \frac{1}{n^\gamma} $$ for some set of $\gamma$, that represent the natural numbers in the 'zeroes' sense? They might be complex numbers for example, or of the form $\gamma = \rho_1^{k_1} \rho_2^{k_2} \cdots$, i.e. in analogy to integers and products of primes.