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Hi:

____________________________________________________________________

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 )

___________________________________________________________________

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

____________________________________________________________________

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 )

___________________________________________________________________

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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