Riemann zeta zeros Fourier like divergent square wave. Can you complete this analogy?
up vote
1
down vote
favorite
The question is to complete this analogy:
$$left|Z(t)right|=left|zeta left(frac{1}{2}+i tright)right| tag{1}$$
is to:
$$Z(t)=e^{i vartheta (t)} zeta left(frac{1}{2}+i tright) tag{2}$$
as:
$$left|f(t)right|=left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i t-1)}}right| tag{3}$$
is to what?
$f(t) = text{?} tag{4}$
I am asking this because $f(t)$ divided by Riemann Siegel theta and the $k$-th Harmonic number:
$$f_{vartheta(t)}(t)=frac{text{sgn}
(Z(t))left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i cdot t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i cdot t-1)}}right|}{g(t)+H_{text{k}}}$$
where:
$$g(t)=frac{partial vartheta (t)}{partial t}$$
and where $vartheta(t)$ is the Riemann-Siegel theta function,
has a nice plot:
The blue graph passes through the Riemann zeta zeros, like the Riemann-Siegel zeta function. However, the square wave $f(t)$ is divergent despite the appearance of the plot.
$Z(t)$ is the Riemann-Siegel zeta function.
number-theory fourier-analysis analytic-number-theory riemann-zeta
asked Nov 17 at 10:08
Mats Granvik
3,34632249
add a comment |
up vote
1
down vote
favorite
The question is to complete this analogy:
$$left|Z(t)right|=left|zeta left(frac{1}{2}+i tright)right| tag{1}$$
is to:
$$Z(t)=e^{i vartheta (t)} zeta left(frac{1}{2}+i tright) tag{2}$$
as:
$$left|f(t)right|=left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i t-1)}}right| tag{3}$$
is to what?
$f(t) = text{?} tag{4}$
I am asking this because $f(t)$ divided by Riemann Siegel theta and the $k$-th Harmonic number:
$$f_{vartheta(t)}(t)=frac{text{sgn}
(Z(t))left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i cdot t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i cdot t-1)}}right|}{g(t)+H_{text{k}}}$$
where:
$$g(t)=frac{partial vartheta (t)}{partial t}$$
and where $vartheta(t)$ is the Riemann-Siegel theta function,
has a nice plot:
The blue graph passes through the Riemann zeta zeros, like the Riemann-Siegel zeta function. However, the square wave $f(t)$ is divergent despite the appearance of the plot.
$Z(t)$ is the Riemann-Siegel zeta function.
number-theory fourier-analysis analytic-number-theory riemann-zeta
asked Nov 17 at 10:08
Mats Granvik
3,34632249
$P_k(s)=sum_{n=1}^k n^{-1} sum_{d |n} mu(d) d^{1-s}$ is a Dirichlet polynomial with real coefficients so $P_k(s) e^{-i h_k(s)/2}$ is real for $Re(s) = 1/2$ where $h_k(s) = log P_k(s) -log P_k(1-s)$ is analytic on $Re(s) = 1/2$.
– reuns
Nov 17 at 10:49
What $f_{vartheta}(t)$ means to you
– reuns
Nov 17 at 10:56
$f_{vartheta}(t)$ means that the function is divided by Riemann Siegelt theta.
– Mats Granvik
Nov 17 at 11:36
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The question is to complete this analogy:
$$left|Z(t)right|=left|zeta left(frac{1}{2}+i tright)right| tag{1}$$
is to:
$$Z(t)=e^{i vartheta (t)} zeta left(frac{1}{2}+i tright) tag{2}$$
as:
$$left|f(t)right|=left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i t-1)}}right| tag{3}$$
is to what?
$f(t) = text{?} tag{4}$
I am asking this because $f(t)$ divided by Riemann Siegel theta and the $k$-th Harmonic number:
$$f_{vartheta(t)}(t)=frac{text{sgn}
(Z(t))left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i cdot t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i cdot t-1)}}right|}{g(t)+H_{text{k}}}$$
where:
$$g(t)=frac{partial vartheta (t)}{partial t}$$
and where $vartheta(t)$ is the Riemann-Siegel theta function,
has a nice plot:
The blue graph passes through the Riemann zeta zeros, like the Riemann-Siegel zeta function. However, the square wave $f(t)$ is divergent despite the appearance of the plot.
$Z(t)$ is the Riemann-Siegel zeta function.
number-theory fourier-analysis analytic-number-theory riemann-zeta
asked Nov 17 at 10:08
Mats Granvik
3,34632249
The question is to complete this analogy:
$$left|Z(t)right|=left|zeta left(frac{1}{2}+i tright)right| tag{1}$$
is to:
$$Z(t)=e^{i vartheta (t)} zeta left(frac{1}{2}+i tright) tag{2}$$
as:
$$left|f(t)right|=left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i t-1)}}right| tag{3}$$
is to what?
$f(t) = text{?} tag{4}$
I am asking this because $f(t)$ divided by Riemann Siegel theta and the $k$-th Harmonic number:
$$f_{vartheta(t)}(t)=frac{text{sgn}
(Z(t))left|sumlimits_{n=1}^{n=k} frac{1}{n} zeta(1/2+i cdot t)sumlimits_{d|n}frac{mu(d)}{d^{(1/2+i cdot t-1)}}right|}{g(t)+H_{text{k}}}$$
where:
$$g(t)=frac{partial vartheta (t)}{partial t}$$
and where $vartheta(t)$ is the Riemann-Siegel theta function,
has a nice plot:
The blue graph passes through the Riemann zeta zeros, like the Riemann-Siegel zeta function. However, the square wave $f(t)$ is divergent despite the appearance of the plot.
$Z(t)$ is the Riemann-Siegel zeta function.
number-theory fourier-analysis analytic-number-theory riemann-zeta
number-theory fourier-analysis analytic-number-theory riemann-zeta
asked Nov 17 at 10:08
Mats Granvik
3,34632249
asked Nov 17 at 10:08
Mats Granvik
3,34632249
edited Nov 17 at 10:13
asked Nov 17 at 10:08
Mats Granvik
3,34632249
asked Nov 17 at 10:08
Mats Granvik
3,34632249
asked Nov 17 at 10:08
Mats Granvik
3,34632249
3,34632249
$P_k(s)=sum_{n=1}^k n^{-1} sum_{d |n} mu(d) d^{1-s}$ is a Dirichlet polynomial with real coefficients so $P_k(s) e^{-i h_k(s)/2}$ is real for $Re(s) = 1/2$ where $h_k(s) = log P_k(s) -log P_k(1-s)$ is analytic on $Re(s) = 1/2$.
– reuns
Nov 17 at 10:49
What $f_{vartheta}(t)$ means to you
– reuns
Nov 17 at 10:56
$f_{vartheta}(t)$ means that the function is divided by Riemann Siegelt theta.
– Mats Granvik
Nov 17 at 11:36
add a comment |
$P_k(s)=sum_{n=1}^k n^{-1} sum_{d |n} mu(d) d^{1-s}$ is a Dirichlet polynomial with real coefficients so $P_k(s) e^{-i h_k(s)/2}$ is real for $Re(s) = 1/2$ where $h_k(s) = log P_k(s) -log P_k(1-s)$ is analytic on $Re(s) = 1/2$.
– reuns
Nov 17 at 10:49
What $f_{vartheta}(t)$ means to you
– reuns
Nov 17 at 10:56
$f_{vartheta}(t)$ means that the function is divided by Riemann Siegelt theta.
– Mats Granvik
Nov 17 at 11:36
$P_k(s)=sum_{n=1}^k n^{-1} sum_{d |n} mu(d) d^{1-s}$ is a Dirichlet polynomial with real coefficients so $P_k(s) e^{-i h_k(s)/2}$ is real for $Re(s) = 1/2$ where $h_k(s) = log P_k(s) -log P_k(1-s)$ is analytic on $Re(s) = 1/2$.
– reuns
Nov 17 at 10:49
$P_k(s)=sum_{n=1}^k n^{-1} sum_{d |n} mu(d) d^{1-s}$ is a Dirichlet polynomial with real coefficients so $P_k(s) e^{-i h_k(s)/2}$ is real for $Re(s) = 1/2$ where $h_k(s) = log P_k(s) -log P_k(1-s)$ is analytic on $Re(s) = 1/2$.
– reuns
Nov 17 at 10:49
What $f_{vartheta}(t)$ means to you
– reuns
Nov 17 at 10:56
What $f_{vartheta}(t)$ means to you
– reuns
Nov 17 at 10:56
$f_{vartheta}(t)$ means that the function is divided by Riemann Siegelt theta.
– Mats Granvik
Nov 17 at 11:36
$f_{vartheta}(t)$ means that the function is divided by Riemann Siegelt theta.
– Mats Granvik
Nov 17 at 11:36
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3002161%2friemann-zeta-zeros-fourier-like-divergent-square-wave-can-you-complete-this-ana%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$P_k(s)=sum_{n=1}^k n^{-1} sum_{d |n} mu(d) d^{1-s}$ is a Dirichlet polynomial with real coefficients so $P_k(s) e^{-i h_k(s)/2}$ is real for $Re(s) = 1/2$ where $h_k(s) = log P_k(s) -log P_k(1-s)$ is analytic on $Re(s) = 1/2$.
– reuns
Nov 17 at 10:49
What $f_{vartheta}(t)$ means to you
– reuns
Nov 17 at 10:56
$f_{vartheta}(t)$ means that the function is divided by Riemann Siegelt theta.
– Mats Granvik
Nov 17 at 11:36