So what happens if there is a non-trivial zero of the Riemann zeta function off the critical line? Has there been any work in the following direction: We know from Landaus theorem that there is a positive proportion of the zeros $\alpha$ on the critical line. Suppose that $\rho$ is a a non-trivial zero of $\zeta$ off the critical line. Then can we use this to cook up an argument to show the existence of another zero off the line? I am being highly optimistic here but continuing in this direction maybe we can show there will be infinite set of zeros off the critical line and going even further we may try to show that the density of this set is greater than $1-\alpha$, yielding the desired contradiction.
[Math] What are the implications of a zero of zeta off the critical line
analytic-number-theorynt.number-theoryriemann-zeta-function
Best Answer
1. First, let us get history right: Hardy (1914) proved there are infinitely many zeros on the critical line, Hardy-Littlewood (1921) proved there are $\gg T$ zeros on the critical line up to height $T$, and Selberg (1942) proved there are $\gg T\log T$ zeros on the critical line up to height $T$ (i.e. positive proportion).
2. I am quite certain that it is not known that if there exists a nontrivial zero off the critical line, then there are infinitely many such zeros.