Ring structures on groups of arithmetic functions

Abstract

Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, 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. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powers f α for α ∈ Bin(R) of a multiplicative arithmetic function f . For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.

The convolution ring of arithmetic functions and symmetric polynomials

“…where λ is listed in weakly decreasing order. For example the Schur polynomial S (3,2) induced by the partition (3,2) in isobaric form is given by…”

“…Rearick used L and E to show that certain groups of arithmetic functions were isomorphic. To our knowledge, there was no follow-up to these definitions and the isomorphism results in the later literature until 2008 [3], where a different version of the operator L is introduced and the isomorphisms are reproved without reference to Rearick's work. We shall extend these definitions to the isobaric ring.…”

“…which after multiplying each term by the appropriate z(α) becomes (3,1) (1 4 ), χ (3,1) (1 2 , 2), χ (3,1) (2 2 ), χ (3,1) (1, 3), χ (3,1)…”

“…In 1968, Rearick [16,17] introduced the notions of "logarithm" and "exponential" functions of arithmetic functions, and used them to prove the isomorphism of M and S, along with a number of other isomorphism of interesting subgroups in A. This result was reproved by Jesse Elliott in 2008 [3] along with a number of other similar results. In 1968 [16,17], Rearick also introduced what he called the trigonometry of numbers by using the properties of the exponential function in the usual way to define "Sine" and "Cosine".…”

“…In 1968 [16,17], Rearick also introduced what he called the trigonometry of numbers by using the properties of the exponential function in the usual way to define "Sine" and "Cosine". In the recent paper of Elliott [3], he does not refer to the inverse operator of the logarithm as an exponential function, but uses his rather more economical, but equivalent definition of logarithm to define its inverse operator. Rearick did not have an explicit expression for his exponential operator, but showed its existence and properties inductively.…”