  Given a collection $\{ G_i\}_{i=1}^d$ of finite groups and a ring $R$, we
have previously introduced and studied certain foundational properties of the
join ring $\mathcal{J}_{G_1, G_2, \ldots, G_d}(R)$. This ring bridges two
extreme worlds: matrix rings $M_n(R)$ on one end, and group rings $R[G]$ on the
other. The construction of this ring was motivated by various problems in graph
theory, network theory, nonlinear dynamics, and neuroscience. In this paper, we
continue our investigations of this ring, focusing more on its arithmetic
properties. We begin by constructing a generalized augmentation map that gives
a structural decomposition of this ring. This decomposition allows us to
compute the zeta function of the join of group rings. We show that the join of
group rings is a natural home for studying the concept of simultaneous
primitive roots for a given set of primes. This concept is related to the order
of the unit group of the join of group rings. Finally, we characterize the join
of group rings over finite fields with the property that the order of every
unit divides a fixed number. Remarkably, Mersenne and Fermat primes
unexpectedly emerge within the context of this exploration.
