caa a22 4500 465824943 CHVBK 20180323112146.0 cr unu---uuuuu 170327e19900201xx s 000 0 eng 10.1007/BF01833941 doi (NATIONALLICENCE)springer-10.1007/BF01833941 Arithmetical identities of the Brauer-Rademacher type [Elektronische Daten] [Pentti Haukkanen, P. McCarthy] 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 α ) = 0 for allp α 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. Birkhäuser Verlag, 1990 Primary 11A25 nationallicence Haukkanen Pentti Department of Mathematical Sciences, University of Tampere, P.O. Box 607, SF-33101, Tampere, Finland aut McCarthy P. Department of Mathematics, University of Kansas, 66045, Lawrence, KS, USA aut aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 40-54 0001-9054 39:1<40 1990 39 10 https://doi.org/10.1007/BF01833941 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01833941 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Haukkanen Pentti Department of Mathematical Sciences, University of Tampere, P.O. Box 607, SF-33101, Tampere, Finland aut NATIONALLICENCE 700 1- McCarthy P. Department of Mathematics, University of Kansas, 66045, Lawrence, KS, USA aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 39/1(1990-02-01), 40-54 0001-9054 39:1<40 1990 39 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer