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The non trivial zeros of Riemann zeta function , x$\zeta(s)$ lies in the critical strip $0<\Re(s)<1$

Riemann Hypothesis states that all the zeros of Riemann zeta function, $\zeta(s)$ lies on the critical line , $\Re(s)=1/2$.

G.H. Hardy proved that an infinity of zeros are on the critical line, $\Re(s)=1/2$

Question Are the number of non trivial zeros of $\zeta(s)$ in the critical strip but not on the critical line finite?

Any help is appreciated.

Angel
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  • https://math.stackexchange.com/questions/1491324/are-the-nontrivial-zeroes-of-the-riemann-zeta-function-countable – jojobo May 11 '21 at 18:30
  • @jojobo Thanks. But I am asking that the zeros not on the critical line but in the strip is finite or not? – Angel May 11 '21 at 18:31
  • In that case, this might help: https://mathoverflow.net/questions/161474/a-couple-of-facts-on-the-non-trivial-zeros-of-the-riemann-zeta-function – jojobo May 11 '21 at 18:32
  • @jojobo Thanks. But answer to the first question in this post is not given – Angel May 11 '21 at 18:35
  • I`m sorry, you could take them as starting points for further research. Also it seems to be an open problem as mentioned in the comments of the second question. – jojobo May 11 '21 at 18:42
  • We don't know if $\sup \Re(\rho)$ is $<1$. – reuns May 11 '21 at 21:18
  • @reuns What do you mean by that? – Angel May 12 '21 at 00:02
  • We know that no zero has real part $1$ but we don't know if a sequence of zeros has real part converging to $1$. – reuns May 12 '21 at 09:48

1 Answers1

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We do not know if there are finitely many ($0$ if RH is true) or infinitely many non-trivial zeros off the critical line. Showing that there are finitely many (not necessarily $0$) would be a huge breakthrough already.

J. W. Tanner
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Gary
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