  The main aim of this paper is to answer certain open questions related to the
exact values of multidimensional Bohr radii by using the concept of arithmetic
Bohr radius for vector-valued holomorphic functions defined in complete
Reinhardt domains in $\mathbb{C}^n$. More precisely, we study the asymptotic
estimates of the arithmetic Bohr radius for holomorphic functions in the unit
ball of $\ell^n_q$ $(1\leq q\leq \infty)$ spaces with values in arbitrary
complex Banach spaces. Many of our results generalize the results obtained by
Defant, Maestre, and Prengel [Q. J. Math. 59, (2008), pp. 189--205].
