# Non trivial zeros of Riemann zeta function

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.

• math.stackexchange.com/questions/1491324/… Commented May 11, 2021 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? Commented May 11, 2021 at 18:31
• In that case, this might help: mathoverflow.net/questions/161474/… Commented May 11, 2021 at 18:32
• @jojobo Thanks. But answer to the first question in this post is not given Commented May 11, 2021 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. Commented May 11, 2021 at 18:42

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.