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
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
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
2 hours 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
2 hours 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
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
edited 47 mins ago
asked 2 hours ago
Feynmanfan85
385
385
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
2 hours 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
2 hours ago
add a comment |
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
2 hours 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
2 hours ago
3
3
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
2 hours ago
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
– Dietrich Burde
2 hours 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
2 hours 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
2 hours ago
add a comment |
1 Answer
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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.
Thanks. Just fixed it.
– William Grannis
2 hours ago
Please see my update, thanks.
– Feynmanfan85
51 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.
Thanks. Just fixed it.
– William Grannis
2 hours ago
Please see my update, thanks.
– Feynmanfan85
51 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.
Thanks. Just fixed it.
– William Grannis
2 hours ago
Please see my update, thanks.
– Feynmanfan85
51 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.
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.
edited 1 hour ago
answered 2 hours ago
William Grannis
960519
960519
Thanks. Just fixed it.
– William Grannis
2 hours ago
Please see my update, thanks.
– Feynmanfan85
51 mins ago
add a comment |
Thanks. Just fixed it.
– William Grannis
2 hours ago
Please see my update, thanks.
– Feynmanfan85
51 mins ago
Thanks. Just fixed it.
– William Grannis
2 hours ago
Thanks. Just fixed it.
– William Grannis
2 hours ago
Please see my update, thanks.
– Feynmanfan85
51 mins ago
Please see my update, thanks.
– Feynmanfan85
51 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
2 hours 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
2 hours ago