Self-intersections of the Riemann zeta function on the critical line

@article{Banks2012SelfintersectionsOT,
  title={Self-intersections of the Riemann zeta function on the critical line},
  author={William D. Banks and Victor Castillo-Garate and Luigi Fontana and Carlo Morpurgo},
  journal={Journal of Mathematical Analysis and Applications},
  year={2012},
  volume={406},
  pages={475-481}
}
Abstract We show that the Riemann zeta function ζ has only countably many self-intersections on the critical line, i.e., for all but countably many z ∈ C the equation ζ ( 1 2 + i t ) = z has at most one solution t ∈ R . More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on R , then either the set { ( a , b ) ∈ R 2 : a ≠ b  and  F ( a ) = F ( b ) } is countable, or the image F ( R ) is a loop in C . 
$a$-Points of the Riemann zeta-function on the critical line
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

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