Continuous and discrete universality of zeta-functions: Two sides of the same coin?
Athanasios Sourmelidis
TL;DR
The paper builds a unified linear-dynamics framework to connect continuous and discrete universality phenomena for zeta- and $L$-functions. By embedding the problem in a Fréchet space of analytic functions and using a translation semigroup, it demonstrates that continuous strong universality for the Hurwitz zeta-function implies its $h$-discrete strong universality for any $h>0$, and conversely under a limsup formulation the discrete case implies the continuous. Under the Riemann hypothesis, continuous universality of the Riemann zeta-function is shown to be equivalent to its $h$-discrete universality, with the Conejero–Müller–Peris framework and Bagchi’s strong recurrence playing central roles; the results extend naturally to multidimensional joint universality and to broader classes of zeta- and $L$-functions, including those without Euler products. The work illuminates why continuous universality suffices to generate a wide array of discrete universality results and provides a robust dynamical toolkit for future exploration of universal approximation in analytic number theory.
Abstract
In 1975 Voronin proved the universality theorem for the Riemann zeta-function $ζ(s)$ which roughly says that any admissible function $f(s)$ is approximated by $ζ(s)$. A few years later Reich proved a discrete analogue of this result. The proofs of these theorems are almost identical but it is not known whether one of them implies the other. We will see that if we translate the question in the language of linear dynamics then there is a link which we exploit to obtain in a straightforward way a big variety of discrete universality results appearing in the literature.