Are there known zeros of the Zeta function off the line 1/2?












4














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







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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
















4














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







share|cite|improve this question




















  • 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














4












4








4







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







share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 49 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














  • 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










1 Answer
1






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6














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.






share|cite|improve this answer























  • Thanks. Just fixed it.
    – William Grannis
    2 hours ago










  • Please see my update, thanks.
    – Feynmanfan85
    53 mins ago











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1 Answer
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1 Answer
1






active

oldest

votes









active

oldest

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active

oldest

votes









6














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.






share|cite|improve this answer























  • Thanks. Just fixed it.
    – William Grannis
    2 hours ago










  • Please see my update, thanks.
    – Feynmanfan85
    53 mins ago
















6














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.






share|cite|improve this answer























  • Thanks. Just fixed it.
    – William Grannis
    2 hours ago










  • Please see my update, thanks.
    – Feynmanfan85
    53 mins ago














6












6








6






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.






share|cite|improve this answer














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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








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
    53 mins ago


















  • Thanks. Just fixed it.
    – William Grannis
    2 hours ago










  • Please see my update, thanks.
    – Feynmanfan85
    53 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
53 mins ago




Please see my update, thanks.
– Feynmanfan85
53 mins ago


















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