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

[José Manuel Rodríguez Caballero]

Dear Gill Williamson,
  Because I was writing too fast, I made a mistake in my submission. Here
is the correction concerning the problem that I think it is undescidable.
Instead of

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


it should be

> Given a list of non-zero integers a_1,...,a_k, if there is n such that
> a_1,...,a_k are the non-zero coefficients of r_2(n, q), return true,
> otherwise, return false.


Kind regards,
Jose M.


On Sat, Jan 29, 2022 at 8:58 PM S. Gill Williamson <
gill.williamson at gmail.com> wrote:

> Hi José,
>
> You might be interested in a "toy model" type approach to a different Clay
> Millennium Problem (attached)
>
> In this approach we create a "toy world" full of generic inputs to subset
> sum target zero. In this toy world, however, P = NP.  But the only known
> proof of the existence of this world seems to require large cardinals.  I
> make no claims about this pushing us closer to the solution of the P vs NP
> problem, but my students and colleagues find it fun to think about (and so
> do I, of course).
>
> It is not inconceivable that some sort of analog exists for your problem
>
> Best wishes,
>
> Gill Williamson
> Professor Emeritus UCSD-CSE
>
>
> On Sat, Jan 29, 2022 at 3:48 PM José Manuel Rodríguez Caballero <
> josephcmac at gmail.com> wrote:
>
>> 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.
>>
>>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20220129/266463be/attachment-0001.html>