# Riemann Zeta function zeros

1. Nov 3, 2009

### MichaelMi

Hi:
____________________________________________________________________

(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

Last edited: Nov 3, 2009
2. Nov 3, 2009

### JG89

OP's post with LaTex fixed:

----------------------------------------------------------------

Hi:
__________________________________________________ __________________

(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 $$\mathit{N} = {\displaystyle \frac {{C_{m}}\,\mathrm{cos}({\rho _{\pi }})}{\sqrt{\pi }}} - {\displaystyle \frac {{C_{p}}\, \mathrm{sin}({\rho _{\pi }})}{\sqrt{\pi }}}$$

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 $$\zeta^{\prime}$$ 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