Re: Disproofs of Riemann Hypothesis

From: *** T. Winter (***.Winter_at_cwi.nl)
Date: 07/25/04 

Date: Sun, 25 Jul 2004 16:41:15 GMT

In article <250720040813352889%edgar@math.ohio-state.edu.invalid> "G. A. Edgar" <edgar@math.ohio-state.edu.invalid> writes:
> In article <ecc5e902.0407250146.183eca44@posting.google.com>, Curioser <janjaweed@excite.com> wrote:
>
> > It is not enough to proclaim zeta(z)=0 where z=0.5+14.134725...i, is
> > it? You must give an unambiguous definition of im(z) and prove that
> > zeta(z)=0. If you can do that then you would have proved that Jiang's
> > proof of Theorem 1 is wrong.
> > Over the years there have been lots of papers (in so-called reputable
> > journals) that have have been devoted to proving that many zeros of
> > zeta lie on the critical line im(z)=1/2. The point is that all these
> > zeros are defined indirectly. Jiang's claim is that all these indirect
> > definitions are false. Can you come up with a clear definition of
> > z=x+iy such that z lies in the critical strip and zeta(z)=0?
>
> There is a nice theorem in complex analysis. To compute the number
> of zeros of an analytic function within a contour, you do a certain
> integral over that contour, divide by 2 pi i, and that's your number.
> When we do it around a small circle surrounding the point 0.5+14.0i
> with radius 1/4, we get the answer 1. There is a zero of zeta
> within that circle. The zeta function can be evaluated numerically
> within a given error, and such a coutour integral can be done
> as accurately as required, and since we know the result of the
> calculation is an integer, we merely need to get the answer close
> enough to tell which integer it is.

Another way is possible (and that is the way how it is shown that there
are so many zeros on the line im(z) = 1/2). The zeta function is continuous
and real valued on the critical line. So it is sufficient to find places
where that function value is negative on that line and other places where
that function value is positive to know that the function has a zero on
that line. 

-- 
*** t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~***/
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