analytic-number-theorycomplex-analysisriemann-hypothesisriemann-zeta
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.
You are going to need a bit of knowledge about complex analysis before you can really follow the answer, but if you start with a function defined as a series, it is frequently possible to extend that function to a much larger part of the complex plane.
For example, if you define $f(x)=1+x+x^2+x^3+...$ then $f$ can be extended to $\mathbb C\setminus \{1\}$ as $g(x)=\frac{1}{1-x}$. Clearly, it is "absurd" to say that $f(2)=-1$, but $g(2)=-1$ makes sense.
The Riemann zeta function is initially defined as a series, but it can be "analytically extended" to $\mathbb C\setminus \{1\}$. The details of this really require complex analysis.
Calculating the non-trivial zeroes of the Riemann zeta function is a whole entire field of mathematics.
From Riemann's Zeta Function, By Harold M. Edwards:
Best Answer
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.