Arithmetical identities of the Brauer-Rademacher type

Summary LetA be a regular arithmetical convolution andk a positive integer. LetA _k (r) = {d: d ^k A(r ^k )}, and letf A _k g denote the convolution of arithmetical functionsf andg with respect toA _k . A pair (f, g) of arithmetical functions is calledadmissible if(f A _k g)(m) 0 for allm and if the functions satisfy an arithmetical functional equation which generalizes the Brauer—Rademacher identity. Necessary and sufficient conditions are found for a pair (f, g) of multiplicative functions to be admissible, and it follows that, if(f A _k g)(m) 0 f(m) for allm, then (f, g) is admissible if and only if itsdual pair (f A _k g, g ^–1 ) is admissible.Iff andg ^–1 areA _k -multiplicative (a condition stronger than being multiplicative), and(f A _k g)(m) 0 for allm, then (f, g) is admissible, calledCohen admissible. Its dual pair is calledSubbarao admissible. If (f A _k g) ^–1 (m) 0 itsinverse pair (g ^–1 , f ^–1 ) is also Cohen admissible.Ifg is a multiplicative function then there exists a multiplicative functionf such that the pair (f, g) is admissible if and only if for everyA _k -primitive prime powerp ⁱ either (i)g(p ⁱ ) 0 or (ii)g(p ) = 0 for allp havingA _k -type equal tot. There is a similar kind of characterization of the multiplicative functions which are first components of admissible pairs of multiplicative functions. IfA _k is not the unitary convolution, then there exist multiplicative functionsg which satisfy (i) and are such that neitherg norg ^–1 isA _k -multiplicative: hence there exist admissible pairs of multiplicative functions which are neither Cohen admissible nor Subbarao admissible.An arithmetical functionf is said to be anA _k -totient if there areA _k -multiplicative functionsf _T andf _V such thatf = f _T A _k f _V ^-1 Iff andg areA _k -totients with(f A _k g)(m) 0 for allm, and iff _V = g _T , then the pair (f, g) is admissible. The class of such admissible pairs includes many pairs which are neither Cohen admissible nor Subbarao admissible. If (f, g) is a pair in this class, and iff(m), (f A _k g) ^–1 (m), g ^–1 (m),f ^–1 (m) andg(m) are all nonzero for allm, then its dual, its inverse, the dual of its inverse, the inverse of its dual and the inverse of the dual of its inverse are also admissible, and in many cases these six pairs are distinct.A number of related results, and many examples, are given.