\(a\)-Points of the Riemann zeta-function on the critical line

Abstract

We investigate the proportion of the nontrivial roots of the equation \(\zeta (s)=a\), which lie on the line \(\Re s=1/2\) for \(a \in \mathbb C\) not equal to zero. We show that at most one-half of these points lie on the line \(\Re s=1/2\). Moreover, assuming a spacing condition on the ordinates of zeros of the Riemann zeta-function, we prove that zero percent of the nontrivial solutions to \(\zeta (s)=a\) lie on the line \(\Re s=1/2\) for any nonzero complex number \(a\).