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The set of multiplicative arithmetic functions over a ring R

(commutative with identity) can be given a unique functorial ring

structure for which (1) the operation of addition is Dirichlet

convolution and (2) multiplication of completely multiplicative

functions coincides with point-wise multiplication. This existence of

this ring structure can be derived from the existence of the ring of

``big'' Witt vectors, and it yields a ring structure on the set of

formal Dirichlet series that are expressible as an Euler product. The

group of additive arithmetic functions over R also has a naturally

defined ring structure, and there is a functorial ring homomorphism

from the ring of multiplicative functions to the ring of additive

functions that is an isomorphism if R is a Q-algebra. An application

is given to zeta functions of schemes of finite type over the ring

of integers.