The Dirichlet Hopf algebra of arithmetics

We study the coalgebraic counterparts of addition and multiplication. This allows to construct two Hopf convolutions, also called Hopf gebras, for both addition and multiplication. Neither of this convolutions is forming a Hopf algebra, however, the multiplicative convolution embodies the Dirichlet convolution of number theoretic functions. There is an opportunity to introduce a new coalgebra structure, called renormalized, such that a nice Hopf algebra structure emerges in such a way that the primitive elements are identical. A subtraction scheme, which might be related to renormalization in quantum field theory, allows to use the nice algebra for computations while actually dealing with the original Hopf convolution. There is a deeper relation of addition and multiplication which relies on n-categories. We give as examples the normal ordering in quantum mechanics and its relation to Stirling numbers and Baxter operators as also the construction of the renormalization coproduct employed in renormalization of quantum fields. An outlook will show how quantum field theoretic methods may be used in number theory.