It is known that there exist infinitely many non-trivial zeros of the Riemann zeta function in the critical strip. Also, we know that infinitely many zeros are on the critical line – more than 1/3 among all, asymptotically – as well as most of the zeros live near the critical line.
In 1984, the theorem of Rademacher and Hlawka showed that the ordinates of the nontrivial zeros of the zeta function are uniformly distributed regardless of the abscissas.
Now, suppose that there exists one non-trivial zero having its real part (say A) other than 1/2. Then, let us call x=A "line A." (the complex plane defined as z=x+iy)
In that case when we take it true,
1) Might infinitely many zeros exist on the line A?
2) If 1) is possible, would they be distributed uniformly?
Best Answer
1) Yes, this is still possible
2) No. We know that they have to distributed with low density, i.e. the number of zeros $z$ with $\Im z < T$ with $\Re z > \sigma >1/2$ is bounded by $T^{4\sigma(1-\sigma) + \epsilon}$ for $\epsilon >0$. So no uniform distribution is possible, since the gaps between consecutive zeros has to grow to infinity.
For sharper results in this direction, I suggest the first chapter of Joern Steuding "Universality of L-functions". This book actually explains pretty good what is going on in the critical stipe off the line $\Re s = 1/2$.