Discussion on "tetra-eta-series" (2007) in MO

[MphLee]

EDIT: Just want to say I know what a characteristic function is. And I know what a radical is. But I didn't think this was a radical, so that's super cool, that this is a radical... lol I only know radicals from Analytic Number Theory, and I def wouldn't of called my function that. Still super cool! Thanks for your comments!!!!!!!

Excuse me... my answer was triggered by your  comment

The function \(\chi_m(n)\) is called something. I can't remember right now. It's on the tip of my tongue...

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Btw I can't claim that this brings more than a equivalent way to formulate the problem. I really have not energy to work on it, as much as I would like. I conjecture that it maybe could based on the reasoning that many number theory problems were solved by translating them into ideal-theoretic language and then straight into algebraic geometry.

The story goes like this. Fix a base ring \(A\) for the rest of our discussion. On one hand we have Ideals, special subsets of rings, while on the other have we have algebraic sets, subsets of some ambient space, think of them as geometric figures whose points can be isolated as solutions of some equation, loci of equations in the language of the fixed ring.

Given a subset \(S\subseteq A[x] \) of ring of polynomial functions we look at the figure they define, as the locus over which they vanish, in the ambient space \({\mathcal V}(S)\subseteq A\).
In the other direction, given a set of points, a figure, in the space \(\Phi \subseteq A\) we ask for the ideal of polynomial functions over it that makes it algebraic... think you are Decartes finding a system of equation that has \(\Phi\) as its solution sets, i.e. that describe that geometric figure: call that ideal \({\mathcal I}(\Phi)\). We get a correspondence going back and forth between ideals of the ring \(A[x]\) and geometric figures, affine sets, of the ambient space \(A\).
\[{\mathcal V}:{\rm ideals}(A[x])
\begin{align}
\longleftarrow\\
\longrightarrow\\
\end{align} {\rm affine\,sets}(A):{\mathcal I}
\]
Those arrows aren't inverse but in some correspondence, Galois correspondence if I recall correctly.. I'm rusty on this, but it is something like they are adjoints.
If you perform on an ideal \(\mathcal V\) followed by \(\mathcal I\) you get the radical operation \(\sqrt: {\rm ideals}(A)\to {\rm ideals}(A)\) ... that is idempotent and a closure operator. But by Grothendieck, we can extend this from polynomial rings to every ring... considering every element of a ring as a kind of function over the spectrum (the Zariski prime spectrum) of that ring, thus defining curves over it.
\[{\mathcal V}:{\rm ideals}(A)
\begin{align}
\longleftarrow\\
\longrightarrow\\
\end{align} {\rm affine\,sets}({\rm spec}(A)):{\mathcal I}
\]
This is very sketchy, I know, and not rigorous on fine details but that mostly the big picture.

The point is... our cases is much more complex than this because we are not working over a ring... but over the monoid of natural numbers under multiplication... so in a sense those sets and your formulas belong to some sort of arithmetic geometry over the natural numbers. When we ask for the radical of a number, and we want to interpret it ideal theoretically we are implicitly thinking of numbers as if they were polynomial functions over a mysterious underlying space... something filling this gap
\[{\mathcal V}:{\rm ideals}(\mathbb N)
\begin{align}
\longleftarrow\\
\longrightarrow\\
\end{align} {\rm affine\,sets}(???):{\mathcal I}
\]
I'm very ignorant... and I don't know exactly how to use all of this. But I know that this is deep and it is related to the passage from integers to the rational numbers... and for arbitrary monoid, e.g. monoid of functions, to integer powers to rational iteration and thus to the discussion we were having in the last thread. If you add that integer iterates carry information on periodic points of a map you have the link to the Artin-Mazur zeta function.

But man.... I think is hard as doing non-commutative geometry... or algebraic geometry over the field of one element, motivic cohomology and that scary stuff of Grothendieck/Konstevich/Connes level.


But I'll stop here, since I do not have anything of value to offer to Gottfried and his problem, again I'm sorry.