  In this work, we develop systematically the ``Dirichlet Hopf algebra of
arithmetics'' by dualizing addition and multiplication maps. We study the
additive and multiplicative antipodal convolutions which fail to give rise to
Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism
axiom. The consequences of the weakened structure, called a Hopf gebra, e.g. on
cohomology are explored. This features multiplicativity versus complete
multiplicativity of number theoretic arithmetic functions. The deficiency of
not being a Hopf algebra is then cured by introducing an `unrenormalized'
coproduct and an `unrenormalized' pairing. It is then argued that exactly the
failure of the homomorphism property (complete multiplicativity) for
non-coprime integers is a blueprint for the problems in quantum field theory
(QFT) leading to the need for renormalization. Renormalization turns out to be
the morphism from the algebraically sound Hopf algebra to the physical and
number theoretically meaningful Hopf gebra. This can be modelled alternatively
by employing Rota-Baxter operators. We stress the need for a
characteristic-free development where possible, to have a sound starting point
for generalizations of the algebraic structures. The last section provides
three key applications: symmetric function theory, quantum (matrix) mechanics,
and the combinatorics of renormalization in QFT which can be discerned as
functorially inherited from the development at the number-theoretic level as
outlined here. Hence the occurrence of number theoretic functions in QFT
becomes natural.
