Let $H(\mathbb{C})$ be the space of holomorphic functions on the complex plane. Then it is well-known that for $a\neq 0$, the translation operator
$$
t\_a(f)\triangleq f(x)\mapsto f(x+a),
$$
is topologically transitive on $H(\mathbb{C})$. Are there known, sufficient conditions for $f$ to by a cyclic vector of this map; i.e. for 
$$
\mathrm{Orb}(f,t\_a)\triangleq \left\{
t\_a^n(f):n \in \mathbb{N}
\right\}$$
to be dense in $H(\mathbb{C})$?



Cyclic vectors of translation operator
Such functions are called universal entire functions. Actually most entire functions
have this property. For 
specific examples, $\zeta$ function has this property.


MR0771576 
Duĭos Ruis,
Universal functions and the structure of the space of entire functions. (Russian)
Dokl. Akad. Nauk SSSR 279 (1984), no. 4, 792–795.


Laurinčikas, A.
The universality of zeta-functions. (English summary)
Proceedings of the Eighth Vilnius Conference on Probability Theory and Mathematical Statistics, Part I (2002).
Acta Appl. Math. 78 (2003), no. 1-3, 251–271.


MR3058520 Gauthier, Paul M. Approximating all meromorphic functions by linear motions of the Riemann zeta-function. Comput. Methods Funct. Theory 12 (2012), no. 2, 517–526. 


