Modulus of Riemann zeta function. [closed]
How to find formula for modulus of Riemann zeta function |$zeta$$(s)|$?. Where:
$$Re(s)>0$$
I am looking for two formulas:
- first shows relation of given modulus to some other functions.
- second shows given modulus as a Dirichlet series.
Is it usefull to use Dirichlet eta function or Euler product?
Thanks in advance for any input.
complex-analysis zeta-functions dirichlet-series
asked Dec 28 '18 at 14:06
mkultramkultra
315
closed as unclear what you're asking by Dietrich Burde, A. Goodier, Saad, Shailesh, Andrew Jan 2 at 4:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
How to find formula for modulus of Riemann zeta function |$zeta$$(s)|$?. Where:
$$Re(s)>0$$
I am looking for two formulas:
- first shows relation of given modulus to some other functions.
- second shows given modulus as a Dirichlet series.
Is it usefull to use Dirichlet eta function or Euler product?
Thanks in advance for any input.
complex-analysis zeta-functions dirichlet-series
asked Dec 28 '18 at 14:06
mkultramkultra
315
closed as unclear what you're asking by Dietrich Burde, A. Goodier, Saad, Shailesh, Andrew Jan 2 at 4:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
"relation to some other functions" does sound a bit vague.
– Dietrich Burde
Dec 28 '18 at 14:08
it thought something like functional equations which contains ( is product of ) some special functions such as Gamma function.
– mkultra
Dec 28 '18 at 16:13
Of course $|zeta(s)|$ can't be a Dirichlet series in $s$ as it is real and non-constant thus non-complex analytic. The best you can get is $$|zeta(sigma+it)|^2=sum_{n=1}^infty n^{-sigma-it} a_n(t), qquad a_n(t)= sum_{d | n} d^{2it}$$ whose analytic continuation is $zeta(s)zeta(s-2it)$. The functional equation for $zeta(s)$ applies obviously to $|zeta(s)|$. What do you hope more ?
– reuns
Dec 28 '18 at 20:01
@reuns How did you obtain the formula?
– Szeto
Dec 29 '18 at 4:29
@Szeto $zeta(sigma+it)zeta(sigma-it) = ...$
– reuns
Dec 29 '18 at 4:55
add a comment |
How to find formula for modulus of Riemann zeta function |$zeta$$(s)|$?. Where:
$$Re(s)>0$$
I am looking for two formulas:
- first shows relation of given modulus to some other functions.
- second shows given modulus as a Dirichlet series.
Is it usefull to use Dirichlet eta function or Euler product?
Thanks in advance for any input.
complex-analysis zeta-functions dirichlet-series
asked Dec 28 '18 at 14:06
mkultramkultra
315
How to find formula for modulus of Riemann zeta function |$zeta$$(s)|$?. Where:
$$Re(s)>0$$
I am looking for two formulas:
- first shows relation of given modulus to some other functions.
- second shows given modulus as a Dirichlet series.
Is it usefull to use Dirichlet eta function or Euler product?
Thanks in advance for any input.
complex-analysis zeta-functions dirichlet-series
complex-analysis zeta-functions dirichlet-series
asked Dec 28 '18 at 14:06
mkultramkultra
315
asked Dec 28 '18 at 14:06
mkultramkultra
315
asked Dec 28 '18 at 14:06
mkultramkultra
315
asked Dec 28 '18 at 14:06
mkultramkultra
315
asked Dec 28 '18 at 14:06
mkultramkultra
315
315
closed as unclear what you're asking by Dietrich Burde, A. Goodier, Saad, Shailesh, Andrew Jan 2 at 4:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Dietrich Burde, A. Goodier, Saad, Shailesh, Andrew Jan 2 at 4:50
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
1
"relation to some other functions" does sound a bit vague.
– Dietrich Burde
Dec 28 '18 at 14:08
it thought something like functional equations which contains ( is product of ) some special functions such as Gamma function.
– mkultra
Dec 28 '18 at 16:13
Of course $|zeta(s)|$ can't be a Dirichlet series in $s$ as it is real and non-constant thus non-complex analytic. The best you can get is $$|zeta(sigma+it)|^2=sum_{n=1}^infty n^{-sigma-it} a_n(t), qquad a_n(t)= sum_{d | n} d^{2it}$$ whose analytic continuation is $zeta(s)zeta(s-2it)$. The functional equation for $zeta(s)$ applies obviously to $|zeta(s)|$. What do you hope more ?
– reuns
Dec 28 '18 at 20:01
@reuns How did you obtain the formula?
– Szeto
Dec 29 '18 at 4:29
@Szeto $zeta(sigma+it)zeta(sigma-it) = ...$
– reuns
Dec 29 '18 at 4:55
add a comment |
1
"relation to some other functions" does sound a bit vague.
– Dietrich Burde
Dec 28 '18 at 14:08
it thought something like functional equations which contains ( is product of ) some special functions such as Gamma function.
– mkultra
Dec 28 '18 at 16:13
Of course $|zeta(s)|$ can't be a Dirichlet series in $s$ as it is real and non-constant thus non-complex analytic. The best you can get is $$|zeta(sigma+it)|^2=sum_{n=1}^infty n^{-sigma-it} a_n(t), qquad a_n(t)= sum_{d | n} d^{2it}$$ whose analytic continuation is $zeta(s)zeta(s-2it)$. The functional equation for $zeta(s)$ applies obviously to $|zeta(s)|$. What do you hope more ?
– reuns
Dec 28 '18 at 20:01
@reuns How did you obtain the formula?
– Szeto
Dec 29 '18 at 4:29
@Szeto $zeta(sigma+it)zeta(sigma-it) = ...$
– reuns
Dec 29 '18 at 4:55
1
1
"relation to some other functions" does sound a bit vague.
– Dietrich Burde
Dec 28 '18 at 14:08
"relation to some other functions" does sound a bit vague.
– Dietrich Burde
Dec 28 '18 at 14:08
it thought something like functional equations which contains ( is product of ) some special functions such as Gamma function.
– mkultra
Dec 28 '18 at 16:13
it thought something like functional equations which contains ( is product of ) some special functions such as Gamma function.
– mkultra
Dec 28 '18 at 16:13
Of course $|zeta(s)|$ can't be a Dirichlet series in $s$ as it is real and non-constant thus non-complex analytic. The best you can get is $$|zeta(sigma+it)|^2=sum_{n=1}^infty n^{-sigma-it} a_n(t), qquad a_n(t)= sum_{d | n} d^{2it}$$ whose analytic continuation is $zeta(s)zeta(s-2it)$. The functional equation for $zeta(s)$ applies obviously to $|zeta(s)|$. What do you hope more ?
– reuns
Dec 28 '18 at 20:01
Of course $|zeta(s)|$ can't be a Dirichlet series in $s$ as it is real and non-constant thus non-complex analytic. The best you can get is $$|zeta(sigma+it)|^2=sum_{n=1}^infty n^{-sigma-it} a_n(t), qquad a_n(t)= sum_{d | n} d^{2it}$$ whose analytic continuation is $zeta(s)zeta(s-2it)$. The functional equation for $zeta(s)$ applies obviously to $|zeta(s)|$. What do you hope more ?
– reuns
Dec 28 '18 at 20:01
@reuns How did you obtain the formula?
– Szeto
Dec 29 '18 at 4:29
@reuns How did you obtain the formula?
– Szeto
Dec 29 '18 at 4:29
@Szeto $zeta(sigma+it)zeta(sigma-it) = ...$
– reuns
Dec 29 '18 at 4:55
@Szeto $zeta(sigma+it)zeta(sigma-it) = ...$
– reuns
Dec 29 '18 at 4:55
add a comment |
0
active
oldest
votes
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
1
"relation to some other functions" does sound a bit vague.
– Dietrich Burde
Dec 28 '18 at 14:08
it thought something like functional equations which contains ( is product of ) some special functions such as Gamma function.
– mkultra
Dec 28 '18 at 16:13
Of course $|zeta(s)|$ can't be a Dirichlet series in $s$ as it is real and non-constant thus non-complex analytic. The best you can get is $$|zeta(sigma+it)|^2=sum_{n=1}^infty n^{-sigma-it} a_n(t), qquad a_n(t)= sum_{d | n} d^{2it}$$ whose analytic continuation is $zeta(s)zeta(s-2it)$. The functional equation for $zeta(s)$ applies obviously to $|zeta(s)|$. What do you hope more ?
– reuns
Dec 28 '18 at 20:01
@reuns How did you obtain the formula?
– Szeto
Dec 29 '18 at 4:29
@Szeto $zeta(sigma+it)zeta(sigma-it) = ...$
– reuns
Dec 29 '18 at 4:55