Re: Disproofs of Riemann Hypothesis

From: Zbigniew Fiedorowicz (fiedorow_at_hotmail.com)
Date: 07/25/04 

Date: Sun, 25 Jul 2004 13:41:05 -0400

*** T. Winter wrote:
> 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.

It's not true that zeta(1/2+it) is a real-valued function of t. However
the functional equation for zeta provides an explicit real-valued
function h(t) such that f(t)=exp(i h(t))*zeta(1/2+it) is real-valued.
One can then find all the roots of zeta on the critical line by looking
for sign changes in f(t). See Chapter 6 of Edwards' book on the zeta
function. While I am not at all an expert in this area, I understand
that the numerics of such calculations are quite delicate. For
instance a flaw in Intel's design of their Pentium I chip was
discovered by somebody doing such calculations.