Recently it became more and more apparent that the study
of holomorphic functions \(f\colon U \rightarrow \mathbb{C}\) on
high dimensional polydiscs (i.e., \(U\) the \(n\)-dimensional
polydisc \(\mathbb{D}^n\) for large \(n\), or the infinite
dimensional polydisc \(\mathbb{D}^\infty\)) is intimately related
with the analytic study of Dirichlet series. Dirichlet series form
a fundamental tool within analytic number theory -- the analytic
theory of the distribution of primes numbers. The aim of this talk
is to comment on some recent developments in this direction mainly
based on joint work with various coauthors: F. Bayart, L. Frerick,
M. Maestre, M. Mastyło, S. Schlüters, and P. Sevilla Peris.
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