Analytic Topology and the Jordan Case

[Ignacio Añón]

Thoughts on these foundational issues in analytic topology, would be very
interesting to me.

We have developed, with a group of researchers, techniques to sophisticate
analytic topology within infinitesimally short, meromorphic, intervals.
Certain weird foundational, set theoretic, issues, came up, and we might
benefit from FOM experts...

Vaughan paratt wrote:

""The Jordan curve theorem, that a simple curve in the plane dissects it
into two connected components, is an even more interesting example...""

In analysis, for instance, even to this day, in 2022, there is no simple
topological proof that the infinite derivatives of an infinitely
differentiable, analytic, regular, entire function, exist: the available
proofs are awkward, ugly, unilluminating, impossible to use concretely to
get analytic continuation around the singularities that destroy
differentiability within meromorphic intervals, nor you can use topology to
calculate residues...

The following is an interesting case, where this topological ignorance is
fatal: the closest people have gotten, to proving Lindelof’s
hypothesis, or Riemann’s
hypothesis, is the so called subconvexity bound: this bound shows
concretely, that the absolute value of the zeta function, within its
meromorphic critical strip, is always less than its imaginary part, when
this part is multiplied e-times and also multiplied by another constant
depending only on e. (e is an infinitesimally small constant).


The smallest value yet found for e, is something like 30/200, and our aim
is of course 0, which would settle Lindelof's hypothesis.

If you talk to the few analysts that have devoted their life to the study
of the zeta function, and who have an intimate bond with the delicacies of
the issues, they would all agree on the following: the reason that it is so
difficult to go beyond this bound, is that the behavior, in infinitesimally
short intervals, of the zeta function, within its meromorphic critical
strip, is faintly understood: it is obvious that these intervals, have some
sort of topology: and yet nothing useful, concrete, can be said about it…

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...


Hence the difficulty in approaching the Lindelof bound, not to

say the Rieman hypothesis horizon. It is not even known, if Lindelof implies
Riemann: to me, it is evident that it does not, but topological ignorance
is blinding analysts...
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