We survey some of the universality properties of the Riemann zeta function
ζ(s) and then explain how to obtain a natural quantization of Voronin's
universality theorem (and of its various extensions). Our work builds on the
theory of complex fractal dimensions for fractal strings developed by the
second author and M. van Frankenhuijsen in \cite{La-vF4}. It also makes an
essential use of the functional analytic framework developed by the authors in
\cite{HerLa1} for rigorously studying the spectral operator a
(mapping the geometry onto the spectrum of generalized fractal strings), and
the associated infinitesimal shift ∂ of the real line:
a=ζ(∂). In the quantization (or operator-valued)
version of the universality theorem for the Riemann zeta function ζ(s)
proposed here, the role played by the complex variable s in the classical
universality theorem is now played by the family of `truncated infinitesimal
shifts' introduced in \cite{HerLa1} to study the invertibility of the spectral
operator in connection with a spectral reformulation of the Riemann hypothesis
as an inverse spectral problem for fractal strings. This latter work provided
an operator-theoretic version of the spectral reformulation obtained by the
second author and H. Maier in \cite{LaMa2}. In the long term, our work (along
with \cite{La5, La6}), is aimed in part at providing a natural quantization of
various aspects of analytic number theory and arithmetic geometry