From: mathwft@math.canterbury.ac.nz (Bill Taylor)
Subject: Re: the Riemann Hypothesis and Halting problems
Date: 17 Feb 2001 03:47:51 GMT
Newsgroups: sci.math
Summary: Elementary statement(s) equivalent to the Riemann Hypothesis

|> > Is the Riemann Hypothesis equivalent to solving the Halting
|> > problem for some Turing machine?
|> 
|> No. Well... The *negation* of the Riemann Hypothesis is equivalent to a
|> particular Diophantine equation being solvable. 

Yes; and in fact it is equivalent to a fairly simple one-quantifier statement
about the integers.  So if RH is false, there is an actual counterexample.

Dunno where I got it from, (someone here is bound to know), but allegedly:-


-----------------------------------------------------------
R.H. is equivalent to the statement:

      For all n    [ SUM(1/k)  -  n^2/2 ]^2  <  36 n^3
      ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

where the SUM is over  k <= d(n)

where   d(x) = PROD[m<x]  PROD[j<=m]  e(j) 

where   e(j) = p for j = p^k   and  1 otherwise.
-----------------------------------------------------------



Hmmmm...  well it's "simple" logically anyway, it's  Pi_0^1  (or is that Pi_1^0?)

And my source (whatever it was) says the first up to 38 values of n look 
really good!  A yawning gap in the inequality, the gaping increasing with  n.
No doubt computer searches are making it look better and better all the time.
RH is becoming more and more true even as we speak...

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           Bill Taylor               W.Taylor@math.canterbury.ac.nz
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  I have a sinking feeling that English has lost the gerund and the gerundive.
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