# Does the Riemann zeta function tell us about the order theoretic properties of the natural numbers?

The classical Möbius function $\mu(n)$ fulfills the multiplicative inversion formula, e.g. see this thread. Now I see in the theory of posets, they generalize the concept of that function, see incidence algebra and from this point of view, the previous definition is the case for the natural numbers and its order induced by divisibility.

In the last link, they also define a "zeta-function" $\zeta(x,y)$ for elements $x,y$ of a poset - I think that's just the boolean version of "$x\le y$" telling me if two numbers $x,y$ are appropriately ordered. There is a formula of how to obtain the inverse of such a $\zeta$ using $\mu$, and this appears to be the source of the equal names.

But I fail to see if this new function is otherwise the analog of the Riemann zeta function $\zeta(z)$ (and other members of the family of continuous zeta functions). Does that analytic function also somehow represent a characteristic function for ordered intervals of numbers? How is the inversion formula a generalization of it, when the number theoretic $\zeta(z)$ doesn't even have two arguments?

Or can I maybe connect $\zeta(x,y)$ to the idea behind the Arithmetic and Dedekind zeta function, which know about the spaces (ideals..) which can be constructed from their domain?

• I am not much familiar with incidence algebra, but my guess is that this $\zeta$ has no connection with Riemann zeta at all. – Balarka Sen Jan 25 '14 at 14:45