  Recently Sourmelidis proved that the discrete universality theorem is
equivalent to the continuous universality theorem for zeta-functions. He treats
both the zero-free universality theorem and the strong universality theorem.
Unfortunately in the zero-free case his result is conditional on a Riemann
hypothesis, and in the strong universality case he only proves the implication
in one direction. We prove this equivalence unconditionally, and also prove an
equivalence with hybrid universality. While the main application of our result
is on Dirichlet series and zeta-functions, our proof is general and does not
use the fact that the functions in question can be represented by Dirichlet
series.

Number Theory;Complex Variables