The ring of number-theoretic functions

Formal power series

“…A proof that Ω is a ring may be modelled in an obvious way on that indicated in [2], The identity element ε of Ω is the function defined by ε(0) = 1, ε(α) = 0 for all vectors a Φ 0. That Ω has no proper divisors of zero will appear in §4 with less trouble than a direct proof at this point.…”

An algebraic independence result for Euler products of finite degree

“…The ring A 1 has been studied from various points of view by a number of authors. We mention in this connection the work of Cashwell and Everett [4], who proved that (A 1 , +, .) is a unique factorization domain.…”

Divisibility of arithmetic functions

“…As is well known [1,2], if / and g are arithmetic functions and we define fg(n) = Σφ f(d)g (n/d) and (/+ S)( n ) -f( n ) + g( n )> the resulting system forms an integral domain D called the Dirichlet algebra. The multiplicative identity δ is defined by δ(l) = 1, δ(n) = 0 if n > I.…”

“…The left side of (6) is δ, and we may rewrite (6) in the form > t/pj if j > 1. By (1) and (3) (b~j) . Therefore each term in the sum in (7) has norm greater than (t/p J ) -p j -t, which contradicts our assumption that the norm of this sum is t. Thus/ (α) = δ, so/is in standard form.…”