Are there known zeros of the Zeta function off the line 1/2?
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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 Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
$endgroup$
add a comment |
$begingroup$
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 Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
$endgroup$
3
$begingroup$
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
$endgroup$
– Dietrich Burde
Dec 28 '18 at 15:42
1
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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 $frac12$.
$endgroup$
– Clayton
Dec 28 '18 at 15:42
$begingroup$
For more on the zeroes and the history of zero finding of the Riemann zeta function, I once geeked out and wrote a fair synopsis.
$endgroup$
– Eric Towers
Dec 28 '18 at 19:17
add a comment |
$begingroup$
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 Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
$endgroup$
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 Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
asked Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
edited Dec 28 '18 at 17:02
Feynmanfan85
asked Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
asked Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
asked Dec 28 '18 at 15:39
Feynmanfan85Feynmanfan85
425
425
3
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$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
$endgroup$
– Dietrich Burde
Dec 28 '18 at 15:42
1
$begingroup$
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 $frac12$.
$endgroup$
– Clayton
Dec 28 '18 at 15:42
$begingroup$
For more on the zeroes and the history of zero finding of the Riemann zeta function, I once geeked out and wrote a fair synopsis.
$endgroup$
– Eric Towers
Dec 28 '18 at 19:17
add a comment |
3
$begingroup$
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
$endgroup$
– Dietrich Burde
Dec 28 '18 at 15:42
1
$begingroup$
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 $frac12$.
$endgroup$
– Clayton
Dec 28 '18 at 15:42
$begingroup$
For more on the zeroes and the history of zero finding of the Riemann zeta function, I once geeked out and wrote a fair synopsis.
$endgroup$
– Eric Towers
Dec 28 '18 at 19:17
3
3
$begingroup$
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
$endgroup$
– Dietrich Burde
Dec 28 '18 at 15:42
$begingroup$
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
$endgroup$
– Dietrich Burde
Dec 28 '18 at 15:42
1
1
$begingroup$
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 $frac12$.
$endgroup$
– Clayton
Dec 28 '18 at 15:42
$begingroup$
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 $frac12$.
$endgroup$
– Clayton
Dec 28 '18 at 15:42
$begingroup$
For more on the zeroes and the history of zero finding of the Riemann zeta function, I once geeked out and wrote a fair synopsis.
$endgroup$
– Eric Towers
Dec 28 '18 at 19:17
$begingroup$
For more on the zeroes and the history of zero finding of the Riemann zeta function, I once geeked out and wrote a fair synopsis.
$endgroup$
– Eric Towers
Dec 28 '18 at 19:17
add a comment |
1 Answer
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oldest
votes
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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 Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
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$begingroup$
Thanks. Just fixed it.
$endgroup$
– William Grannis
Dec 28 '18 at 15:49
$begingroup$
Please see my update, thanks.
$endgroup$
– Feynmanfan85
Dec 28 '18 at 16:57
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
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 Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
$endgroup$
$begingroup$
Thanks. Just fixed it.
$endgroup$
– William Grannis
Dec 28 '18 at 15:49
$begingroup$
Please see my update, thanks.
$endgroup$
– Feynmanfan85
Dec 28 '18 at 16:57
add a comment |
$begingroup$
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 Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
$endgroup$
$begingroup$
Thanks. Just fixed it.
$endgroup$
– William Grannis
Dec 28 '18 at 15:49
$begingroup$
Please see my update, thanks.
$endgroup$
– Feynmanfan85
Dec 28 '18 at 16:57
add a comment |
$begingroup$
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 Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
$endgroup$
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 Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
edited Dec 28 '18 at 21:04
answered Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
answered Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
answered Dec 28 '18 at 15:42
William GrannisWilliam Grannis
1,008521
1,008521
$begingroup$
Thanks. Just fixed it.
$endgroup$
– William Grannis
Dec 28 '18 at 15:49
$begingroup$
Please see my update, thanks.
$endgroup$
– Feynmanfan85
Dec 28 '18 at 16:57
add a comment |
$begingroup$
Thanks. Just fixed it.
$endgroup$
– William Grannis
Dec 28 '18 at 15:49
$begingroup$
Please see my update, thanks.
$endgroup$
– Feynmanfan85
Dec 28 '18 at 16:57
$begingroup$
Thanks. Just fixed it.
$endgroup$
– William Grannis
Dec 28 '18 at 15:49
$begingroup$
Thanks. Just fixed it.
$endgroup$
– William Grannis
Dec 28 '18 at 15:49
$begingroup$
Please see my update, thanks.
$endgroup$
– Feynmanfan85
Dec 28 '18 at 16:57
$begingroup$
Please see my update, thanks.
$endgroup$
– Feynmanfan85
Dec 28 '18 at 16:57
add a comment |
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$begingroup$
$zeta(-2n)=0$ are "trivial" zeros off the line $1/2$, see this post.
$endgroup$
– Dietrich Burde
Dec 28 '18 at 15:42
1
$begingroup$
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 $frac12$.
$endgroup$
– Clayton
Dec 28 '18 at 15:42
$begingroup$
For more on the zeroes and the history of zero finding of the Riemann zeta function, I once geeked out and wrote a fair synopsis.
$endgroup$
– Eric Towers
Dec 28 '18 at 19:17