Prev: Flight Tickets Booking Chicago to Mumbai | Cheap Airfares to India Destinations
Next: But who trusts Usenet?
From: Cliff Bott on 28 Mar 2010 07:52
Perhaps you meant to ask whether there are any 'non-trivial' zeros of the
zeta function.

The 'trivial' zeros of the zeta function are related to the poles of the
gamma function through Riemann's 'functional equation'.

The first few 'non-trivial' zeros were calculated by Riemann who confirmed
that they have real part 1/2 (and thus conform with the 'Riemann
hypothesis').

Edwards' book on the zeta function has a good discussion of Riemann's
exploration of the properties of the zeta function in his profound 1859
paper but is not mathematically rigorous. Whittaker and Watson gives a good
and mathematically rigorous discussion of the properties of the gamma and
zeta functions and their inter-relationship.

"Robert Adams" <robert.adams(a)analog.com> wrote in message
news:4765c269-4117-4976-a1dc-cbadd1b6e7fa(a)x11g2000prb.googlegroups.com...
> Are there any "non-trvial" poles of the zeta function? If so, have
> they been tabulated?
>
> Looking at the Hadamard product form, it looks like the poles would be
> equal to the zeros of gamma(1 + s/2).
>
>
> Bob


From: Robert Adams on 28 Mar 2010 08:17
On Mar 28, 7:52 am, "Cliff Bott" <cliff_b...(a)bigpond.com> wrote:
> Perhaps you meant to ask whether there are any 'non-trivial' zeros of the
> zeta function.
>
> The 'trivial' zeros of the zeta function are related to the poles of the
> gamma function through Riemann's 'functional equation'.
>
> The first few 'non-trivial' zeros were calculated by Riemann who confirmed
> that they have real part 1/2 (and thus conform with the 'Riemann
> hypothesis').
>
> Edwards' book on the zeta function has a good discussion of Riemann's
> exploration of the properties of the zeta function in his profound 1859
> paper but is not mathematically rigorous. Whittaker and Watson gives a good
> and mathematically rigorous discussion of the properties of the gamma and
> zeta functions and their inter-relationship.
>
> "Robert Adams" <robert.ad...(a)analog.com> wrote in message
>
> news:4765c269-4117-4976-a1dc-cbadd1b6e7fa(a)x11g2000prb.googlegroups.com...
>
>
>
> > Are there any "non-trvial" poles of the zeta function? If so, have
> > they been tabulated?
>
> > Looking at the Hadamard product form, it looks like the poles would be
> > equal to the zeros of gamma(1 + s/2).
>
> > Bob- Hide quoted text -
>
> - Show quoted text -

No, I meant to ask about poles, not zeros. The reason I ask is that I
fed the magnitude of the zeta function on the re=1/2 line into an
electrical engineering program that attempts to match magnitude
functions by fitting it with poles and zeros, and it seems to come up
with an exact match. Since it is only looking at the complex
magnitude, it is unlikely that it is matching both real and imaginary
parts, but still I find it somewhat intruiging. I assign a pole for
each zero and constrain it to have the same imaginary part as its
associated zero, but let the iterative program decide on the real
part. The real part of the pole controls the "width" of the notch
function around the zero.
Since I am an eletrical engineer, my only goal here is to define a
dynamical system (circuit) that has a "frequency response" that
matches the zeta function magnitude on the critical line. So far it
looks like it should be possible. Probably will result only in a
laboratory curiosity, but at least it's different, and fun!
First  |  Prev  | 
Pages: 1 2
Prev: Flight Tickets Booking Chicago to Mumbai | Cheap Airfares to India Destinations
Next: But who trusts Usenet?