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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.

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  • $\begingroup$ math.stackexchange.com/questions/1491324/… $\endgroup$
    – jojobo
    Commented May 11, 2021 at 18:30
  • $\begingroup$ @jojobo Thanks. But I am asking that the zeros not on the critical line but in the strip is finite or not? $\endgroup$
    – Angel
    Commented May 11, 2021 at 18:31
  • $\begingroup$ In that case, this might help: mathoverflow.net/questions/161474/… $\endgroup$
    – jojobo
    Commented May 11, 2021 at 18:32
  • $\begingroup$ @jojobo Thanks. But answer to the first question in this post is not given $\endgroup$
    – Angel
    Commented May 11, 2021 at 18:35
  • $\begingroup$ 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. $\endgroup$
    – jojobo
    Commented May 11, 2021 at 18:42

1 Answer 1

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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.

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  • $\begingroup$ Thanks for the answer $\endgroup$
    – Angel
    Commented May 11, 2021 at 19:43

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