Analytic Topology and the Jordan Case (Ignacio A??n)

[José Manuel Rodríguez Caballero]

Ignacio A??n

> The few known interesting facts about the topology of the infinitesimally
> short, meromorphic intervals of the zeta function, are derived from
> exponential sums, trigonometric sums, sophistications of the
> Hardy-Littlewood method, and Hadamard's three circle theorem; they are
> rather crude substitutes for topological knowledge, or for analytic
> techniques like Jensen's theorem, or Nevanlinna theory. These analytic
> techniques, give precise control over the value distribution of meromorphic
> intervals, but are difficult to use in the zeta function, since topological
> knowledge about the invariant particularities of the zeta function, is
> missing...


I think that the relevant question for foundations of mathematics is
whether ZFC is powerful enough to deduce either the proof of the Riemann
hypothesis or the proof of a counterexample. A first step would be to
develop an undecidable toy model analog of the Riemann hypothesis.

Some analogs of the Riemann hypothesis have been developed over finite
fields. They are easier to understand than the original Riemann zeta
funciton. Alain Connes's program [1] is to go from a field of
characteristic q to a (hypothetical) field of characteristic 1. He uses the
formalism of Topos Theory to define his field of characteristics 1.

In my own research [2], I have developed a q-analog of Jacobi's formula for
the number of representations of an integer as the sum of two squares,
denoted r_2(n, q). I suspect that the following problem is undecidable:

Given a polynomial as input P(q), if there is n such that r_2(n, q) = P(q),
return true, otherwise, return false.

If this problem is decidable, I would appreciate seeing the algorithm. If
it is undecidable, the function that I used to define r_2(n, q) may give
some clue to prove undecidability in areas of complex analysis with a
strong connection to number theory. This function is a variation (change of
sign) of the Jordan-Kronecker function [3].

Kind regards,
Jose M.

[1] Connes, Alain, Caterina Consani, and Matilde Marcolli. "Fun with
F1." *Journal
of Number Theory* 129.6 (2009): 1532-1561.
https://www.sciencedirect.com/science/article/pii/S0022314X08001856

[2] A q-analog of Jacobi's two squares formula and its applications
https://arxiv.org/pdf/1801.03134.pdf

[3]  Shaun Cooper. Ramanujan’s theta functions. Springer, 2017.
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