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In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:

Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to independence results for publication.

Question 1. can someone give a short description of the stated work(s)?

Question 2. Do such independence results say anything about the independence of Riemann hypothesis or ABC conjecture in $PA$ or some of its weaker sub-theories?

The paper Unprovability, phase transitions and the Riemann zeta-function by Bovykin-Wiermann may be related. Also as Jaso Rute has suggested, the paper Phase transitions in logic and combinatorics might be also helpful.

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    $\begingroup$ I have applied a protection to prevent an answer being posted by a certain repeat offender whom I shall not name, but who posts spammy answers that he claims have to do with the Riemann Hypothesis and ABC conjecture, whether or not they are relevant to the OP. $\endgroup$
    – Todd Trimble
    Commented Dec 20, 2014 at 14:12
  • $\begingroup$ @Todd: Thanks in advance! (And in retrospect, I suppose.) $\endgroup$
    – Asaf Karagila ♦
    Commented Dec 20, 2014 at 14:15
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    $\begingroup$ Why don't you ask Andreas himself? A simple email would do. $\endgroup$
    – user1688
    Commented Dec 20, 2014 at 15:45
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    $\begingroup$ You may also want to look at this paper: cage.ugent.be/~weierman/MSJ.pdf . See the first paragraph on page 3, although the whole paper looks quite readable at a quick glance. $\endgroup$
    – Jason Rute
    Commented Dec 20, 2014 at 16:16
  • $\begingroup$ Is there a link for the preprint? $\endgroup$
    – joro
    Commented Dec 20, 2014 at 17:42

1 Answer 1

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Presumably this is my task:

Question 1. can someone give a short description of the stated work(s)?

The work centers around the "phase transition for Gödel incompleteness" program. The basis idea is to consider an assertion $A(f)$ depending on a function parameter $f$ so that $A(f)$ is true, $A(f)$ is PA-provable if $f$ is slow growing and $A(f)$ is PA-unprovable if $f$ is fast growing. We assume that $A(f)$ is monotone in $f$ with respect to unprovability. The goal is to classify the threshold region where the transition from provability to unprovability happens. The standard examples for $A(f)$ stem from Ramsey theory, wqo-theory and the theory of well-orders. Quite often analytic combinatorics (Tauberians, etc) has to be combined with proof theory to get decent results. The idea in using unproven hypotheses in this business is to sharpen bounds on the threshold region in some meaningful way. RH affects the numbers of primes in short intervals and ABC affects the number of square free numbers in short intervals. This allows in specific contexts to refine the threshold window. The drawback is that the resulting assertions $A(f)$ do not look very natural from the viewpoint of logic (and the referee was not too enthousiastic then).

Question 2. Do such independence results say anything about the independence of Riemann hypothesis or ABC conjecture in PA or some of its weaker sub-theories?

This research does not say much about the independence of the hypotheses in PA. Studying $A(f)$ could in principle be used to disprove hypotheses used for refining the threshold region. But this would be more sensible for hypotheses which are assumed to be false.

The joint work with Andrey Bovykin is of a different nature. It offers a perspective to incorporate results on the value distribution of the $\zeta$-function to prove independence results. The basis idea is to model Ramseyan statements.

Andrey has also some interesting ideas to get unprovability of some unproven number-theoretic hypotheses in PA. So he might be asked for further comments.

Best, Andreas

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