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Some nice PA incompleteness
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To start my FMPC contribution this year I cover some recent
results on PA and fragments.

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Motivated by recent work by A. Bovykin on Friedman's sine principle (which
seems to be of ground breaking importance)
I looked at the following assertion concerning Riemann's zeta function.

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Zeta(sigma):= For all n and for all K there exists R so large such that 
for all rational numbers
r_1<,...,r_R we find a set H a subset of {1,....,R} of cardinality K 
such that<BR>
for all increasing sequences h_1<h_2<...<h_K
and h_1<h'_2<...<h'_K
the absolute value of
\zeta(sigma+i.r_{h_1}.r_{h_2}.r_{h_K}-\zeta(sigma+i.r_{h_1}.r_{h'_2}.r_{h'_K}
is smaller than 2^{-h_1}

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(Here sigma is a prim. recursive real number.)

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This principle Zeta(sigma) is true but unprovable in PA for all sigma 
different from 0.5.

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If sigma=0.5 a similar result will hold for a suitably normalized zeta.
Here Chris Hughes, a zeta expert, provided a critical estimate.
(RH will not be required)

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Explanation: Kanamori McAloon is hidden in the formulations but in a not too
obvious way. Further I used Andrey's approach (paper to appear in PAMS)
plus results on simultaneous diophantine approximation.


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Going a bit further one can prove the following. It is true but 
unprovable in I\Sigma_1 that:<BR>
For all K there exists R so large that for all choices of natural 
numbers m_1<,...,m_R we find
a subset H of cardinality K such that for all k,l,m in H with k<l  and  
k<m we have that<BR>
\mid  \zeta^{(k)}(0.75+i\cdot l)  -   \zeta^{(k)}(0.75+i\cdot m)      
\mid  < \frac 1 {2^k}<BR>
where \zeta^(k) denotes the k-th derivative.

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I believe that such results hold for all sorts of Dirichlet series 
showing up in cutting edge math.
(L-series, zeta functions of number fields, other zeta functions.)  
Proving this will require extension
of current knowledge of these functions (maybe a nice PhD project, maybe 
a bit more).

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Further principles will involve the spacing of zeta zeros and
similar results will hold too. Another direction which I looked at
is ergodic theory. One can replace \zeta in a suitable way
by iterations of  the quadratic
map or the Gauss map to obtain independences too. I expect here
results for the geodesic flow on the hyperbolic plane too.
Also random matrix theory might be integrated.

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The whole subject turns out to be very rich and promising.
I hope that logicians find this appealing and that
pure mathematicians may find it interesting, too.

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Andreas Weiermann