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Added Nov.3, 2009

(For anyone who can't read the formula below (probably everyone) and who

might have an interest in the subject: - the derivation of two simple equations

that locate all the zeros of the zeta function on the imaginary (critical) line can be downloaded from

http://www.magma.ca/~gmtrcs/papers/zeta.pdf )

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Can anyone tell me if the formula below is already known?

The zeros of the Zeta function along the imaginary (critical) line coincide with the zeros of

the following equation:

{D_{R}\,{\zeta _{I}^{\prime} } } + {N}\,{{\zeta _{R}^{\prime} } =0 }.

where N =

\mathit{N} = {\displaystyle \frac {{C_{m}}\,\mathrm{cos}({\rho

_{\pi }})}{\sqrt{\pi }}} - {\displaystyle \frac {{C_{p}}\,

\mathrm{sin}({\rho _{\pi }})}{\sqrt{\pi }}}

and D is

\mathit{D_{R}} = {\displaystyle \frac {1}{2}} - {\displaystyle

\frac {1}{2}} \,{\displaystyle \frac { {C_{p}}\,\mathrm{cos}({

\rho _{\pi }}) + {C_{m}}\,\mathrm{sin}({\rho _{\pi }})}{\sqrt{\pi

}}}

{C_{p}}& =& \mathrm{cosh}({\displaystyle \frac {\pi \,\rho }{2}} )\,{

\Gamma _{R}} + \mathrm{sinh}({\displaystyle \frac {\pi \,\rho }{2

}} )\,{\Gamma _{I}}\\

{C_{m}}& = & - \mathrm{sinh}({\displaystyle \frac {\pi \,\rho }{2}} )

\,{\Gamma _{R}} + \mathrm{cosh}({\displaystyle \frac {\pi \,\rho

}{2}} )\,{\Gamma _{I}}

Gamma _{I} is the imaginary part of Gamma(1/2+I*rho)

Gamma _{R} is the Real part of Gamma(1/2+I*rho)

and similarly for Zeta\prime, the first derivative of Zeta (s)

with s=1/2+I*rho

I am new to this forum and it does not seem possible to attach a file to this message.

Or of it is, it doesn't seem to work for me.

If someone would like a copy of the derivation of this formula, please send a message with

an email address and I will send a copy of the full derivation.

Thank you

Mike

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# Riemann Zeta function zeros

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