We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.