Are there known zeros of the Zeta function off the line 1/2?
I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers.
In short, we'd be looking at the Zeta function using values of s produced by powers of the logarithm function:
$zeta(log^n (-1)/x).$
I noticed several zeros for n = 6:
https://www.wolframalpha.com/input/?i=plot+zeta(log%5E6(-1)%2Fx),+x+from+1+to+500
But, I don't know enough about this topic to be sure that this is actually interesting, since I know that at least some of the zeros of this function are considered "trivial". Nonetheless, I thought I'd share this in case some of you found it interesting and had insights.
riemann-zeta zeta-functions riemann-hypothesis
asked 1 hour ago
Feynmanfan85
285
add a comment |
I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers.
In short, we'd be looking at the Zeta function using values of s produced by powers of the logarithm function:
$zeta(log^n (-1)/x).$
I noticed several zeros for n = 6:
https://www.wolframalpha.com/input/?i=plot+zeta(log%5E6(-1)%2Fx),+x+from+1+to+500
But, I don't know enough about this topic to be sure that this is actually interesting, since I know that at least some of the zeros of this function are considered "trivial". Nonetheless, I thought I'd share this in case some of you found it interesting and had insights.
riemann-zeta zeta-functions riemann-hypothesis
asked 1 hour ago
Feynmanfan85
285
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
1 hour ago
1
There are no known zeros of $zeta(s)$ that don’t occur at a negative even integer (these are the trivial zeros) or have real part $sigma=frac12$. The Riemann Hypothesis states that any zero with real part between $0$ and $1$ will in fact have real part exactly $frac12$.
– Clayton
1 hour ago
add a comment |
I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers.
In short, we'd be looking at the Zeta function using values of s produced by powers of the logarithm function:
$zeta(log^n (-1)/x).$
I noticed several zeros for n = 6:
https://www.wolframalpha.com/input/?i=plot+zeta(log%5E6(-1)%2Fx),+x+from+1+to+500
But, I don't know enough about this topic to be sure that this is actually interesting, since I know that at least some of the zeros of this function are considered "trivial". Nonetheless, I thought I'd share this in case some of you found it interesting and had insights.
riemann-zeta zeta-functions riemann-hypothesis
asked 1 hour ago
Feynmanfan85
285
I've been doing research in information theory that had some superficial connections to the Zeta function, since I was making use of logarithms of complex numbers.
In short, we'd be looking at the Zeta function using values of s produced by powers of the logarithm function:
$zeta(log^n (-1)/x).$
I noticed several zeros for n = 6:
https://www.wolframalpha.com/input/?i=plot+zeta(log%5E6(-1)%2Fx),+x+from+1+to+500
But, I don't know enough about this topic to be sure that this is actually interesting, since I know that at least some of the zeros of this function are considered "trivial". Nonetheless, I thought I'd share this in case some of you found it interesting and had insights.
riemann-zeta zeta-functions riemann-hypothesis
riemann-zeta zeta-functions riemann-hypothesis
asked 1 hour ago
Feynmanfan85
285
asked 1 hour ago
Feynmanfan85
285
edited 23 mins ago
asked 1 hour ago
Feynmanfan85
285
asked 1 hour ago
Feynmanfan85
285
asked 1 hour ago
Feynmanfan85
285
285
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
1 hour ago
1
There are no known zeros of $zeta(s)$ that don’t occur at a negative even integer (these are the trivial zeros) or have real part $sigma=frac12$. The Riemann Hypothesis states that any zero with real part between $0$ and $1$ will in fact have real part exactly $frac12$.
– Clayton
1 hour ago
add a comment |
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
1 hour ago
1
There are no known zeros of $zeta(s)$ that don’t occur at a negative even integer (these are the trivial zeros) or have real part $sigma=frac12$. The Riemann Hypothesis states that any zero with real part between $0$ and $1$ will in fact have real part exactly $frac12$.
– Clayton
1 hour ago
3
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
1 hour ago
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
1 hour ago
1
1
There are no known zeros of $zeta(s)$ that don’t occur at a negative even integer (these are the trivial zeros) or have real part $sigma=frac12$. The Riemann Hypothesis states that any zero with real part between $0$ and $1$ will in fact have real part exactly $frac12$.
– Clayton
1 hour ago
There are no known zeros of $zeta(s)$ that don’t occur at a negative even integer (these are the trivial zeros) or have real part $sigma=frac12$. The Riemann Hypothesis states that any zero with real part between $0$ and $1$ will in fact have real part exactly $frac12$.
– Clayton
1 hour ago
add a comment |
1 Answer
1
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oldest
votes
First of all, when plotted, only the real part of your graph crosses zero. Secondly, the only known zeroes of the zeta function off of the ${1 over 2} + atimes i$ line are the negative even integers, each of which is a zero. These are called trivial zeroes. The Riemann Hypothesis states that all other zeroes lie on that magic line. It is known that at least 40% of them lie there, and none off of it have ever been found.
answered 1 hour ago
William Grannis
885519
Thanks. Just fixed it.
– William Grannis
1 hour ago
Please see my update, thanks.
– Feynmanfan85
28 mins ago
add a comment |
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First of all, when plotted, only the real part of your graph crosses zero. Secondly, the only known zeroes of the zeta function off of the ${1 over 2} + atimes i$ line are the negative even integers, each of which is a zero. These are called trivial zeroes. The Riemann Hypothesis states that all other zeroes lie on that magic line. It is known that at least 40% of them lie there, and none off of it have ever been found.
answered 1 hour ago
William Grannis
885519
Thanks. Just fixed it.
– William Grannis
1 hour ago
Please see my update, thanks.
– Feynmanfan85
28 mins ago
add a comment |
First of all, when plotted, only the real part of your graph crosses zero. Secondly, the only known zeroes of the zeta function off of the ${1 over 2} + atimes i$ line are the negative even integers, each of which is a zero. These are called trivial zeroes. The Riemann Hypothesis states that all other zeroes lie on that magic line. It is known that at least 40% of them lie there, and none off of it have ever been found.
answered 1 hour ago
William Grannis
885519
Thanks. Just fixed it.
– William Grannis
1 hour ago
Please see my update, thanks.
– Feynmanfan85
28 mins ago
add a comment |
First of all, when plotted, only the real part of your graph crosses zero. Secondly, the only known zeroes of the zeta function off of the ${1 over 2} + atimes i$ line are the negative even integers, each of which is a zero. These are called trivial zeroes. The Riemann Hypothesis states that all other zeroes lie on that magic line. It is known that at least 40% of them lie there, and none off of it have ever been found.
answered 1 hour ago
William Grannis
885519
First of all, when plotted, only the real part of your graph crosses zero. Secondly, the only known zeroes of the zeta function off of the ${1 over 2} + atimes i$ line are the negative even integers, each of which is a zero. These are called trivial zeroes. The Riemann Hypothesis states that all other zeroes lie on that magic line. It is known that at least 40% of them lie there, and none off of it have ever been found.
answered 1 hour ago
William Grannis
885519
edited 41 mins ago
answered 1 hour ago
William Grannis
885519
answered 1 hour ago
William Grannis
885519
answered 1 hour ago
William Grannis
885519
885519
Thanks. Just fixed it.
– William Grannis
1 hour ago
Please see my update, thanks.
– Feynmanfan85
28 mins ago
add a comment |
Thanks. Just fixed it.
– William Grannis
1 hour ago
Please see my update, thanks.
– Feynmanfan85
28 mins ago
Thanks. Just fixed it.
– William Grannis
1 hour ago
Thanks. Just fixed it.
– William Grannis
1 hour ago
Please see my update, thanks.
– Feynmanfan85
28 mins ago
Please see my update, thanks.
– Feynmanfan85
28 mins ago
add a comment |
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3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
1 hour ago
1
There are no known zeros of $zeta(s)$ that don’t occur at a negative even integer (these are the trivial zeros) or have real part $sigma=frac12$. The Riemann Hypothesis states that any zero with real part between $0$ and $1$ will in fact have real part exactly $frac12$.
– Clayton
1 hour ago