Revista Clmbiana de Matematias Vlumen II,1968,pags.6-11 A NOTE ON GENERALIZED MOBIUS f-functions V.S. by ALBIS In [1] the ncept f a cnjugate pair f sets f psi tive integers is int~dued Briefly, if Z dentes the set f psitive integers and P and Q dente nn-empty subsets f Z such that. if n l E Z, n 2 E Z, (n l,n 2 ) 1, then (1) n = n l n 2 E P(resp.Q) <=> n l EP,n 2 EP (resp. Q), and, if in additin, fr each integer n E Z there is a unique factrizatin f the frm (2) n = ab, a E P, b E Q, we say that each f the sets P and Q is a direct factr set f Z, and that (p,q) is a cnjugate pair. It is clear that pnq = {11. Amng the generalized functins studied in [lj, we find (3) pp(n) = E ~(n/d) dep a generalizatin results are als prved in [1]: f MOBIUS Jl-functin. The fllwing (i) pp is a multiplicative functin. E fjp(n/d) if n = 1 if n > 1 6
Here we shall shw that flp is the unique arithmetical functin satisfying (ii) abve. Let r- be such that -f: if n > 1 (4 ) I:,...11 (n/d) = p (n) if n = 1 If ~* is multiplicative, it suffices t prve that ~p(p ) = ~~(pk) fr every prime p and every integer k> O. S let ~* be a multiplicative functin; it fllws frm (4) and (ii) that Jlp(l) = Jlll' (1), thus ~(p) = p*(p) fr every prime p. We will nw shw by inductin n k that ~(pk) = r~(pk). Suppse this relatin hlds fr u > k > O. Frm (ii) we btain because tain (5) ~(pu) = _ ~ pp(pu/pi) l<l.<u pie"q 1 = P E Q. On the ther hand, frm ( 6) _ I: J1* (pu/pi) l<i<u re"q because 1 = p 0 E Q. But by the inductin hypthesis, f1p(pu-i) = Jlt(pU-i) ( i = 1,, u) Thus the rigth members in (5) and (6) are equal, s that Jip(pU) = fl~(plj.) In view f the abve result, it suffices t shw that any functin p~ satisfying (4) is multiplicative, thus prving the fllwing 'rg: QB!2~Ll. If J1~ satisfies (4) fr every nez, 4 = J1p then... 7
In rder t prve this therem we begin with sme lemmata. ([2})!:~~~~L1 Let e be a multiplicative functin. If gel) = 0 => g(n) = 0 fr every n E Z. If gel) I 0 => gel) = 1.!:~~~g. Let f be an arithmetical functin. If Edln, fen/d) 0 fr every nez, then fen) = 0 fr every n E Z. Prf: As in [1, lemma 2], we prceed by indutin n n, nting that fen) always appears in the invlved sum, beca.use 1 E Q. 1~~~2. Let g be a multipliative functin. If f is suh that g(n) E fen/d) then f is multiplicative. Prf: The prf is a cnvenient and trivial adaptatin f that f lemma 3, [2]. Fr the sake f clearness we repeat it. If gel) = 0, then g(n) = 0 fr every nez, s by lemma 2, fen) = 0 fr every n E Z. If gel) ~ 1 it is clear that that f(l) = 1. Let us cnsider the fllwing prpsitin: P t m,n' : «(m,n )=1 => f (mn) = f (m)f (n) >> If m r n = 1, the abve prpsitin is true. Let us suppse that Pfm,n\ is false fr sme pair (m,n), and let m min {m; 3: nez with (m,n)=l suh that P(m,n\is false} 8
then there exists an n such that Pfm,n\ is false. Nw let (m,n) = 1 and n = min{n; (m,n)=l and We have then: i) 1 < m < n and (m,n ) 1-0 0 ii) plm,n \ is false. iii) Plk,nl is true fr every n and each k such that 1 < k < m and (k,n) = 1. - 0 iv) Pfm,tl is true fr each t such that 1 < t < n 0 0 and (m,t) = 1. 0 If nw we take gem n) we find L f(m nit) = gem n ) = gem )g(n ) t ' 1m n 0 0 0 0 0 0 teq = E f(m /d),. dim 0 E fen 10) 0\ n 0 EQ using the multiplicativity f g and (1); s that dim E [f(m n Id),ln d,oeq - f(m Id)f(n l)} = But m Id and n 10 are smaller than m and n 0 0 0 0 resp., if d, f 1; thus frm the abve relatin and the hypthesis n (m,n ), we cnclude that f(m n ) - f(m )f(n ) = 0, 0 0 cntradicting ii). S Plm,n\ is always true. We remark that n explicit calculatin fr pp is 9
is needed in the abve reasnings. Furtlier, if we use the fllwing therem [1, therem ~: ~g~qb M_g.If f and g are arithmetical functins, then (iii) g(n) = E the uniqueness f ~ ved Therem 1 withut fen/d) <=> fen) = l:g(d)f-lp(n/d)., is easily prved. Here we have prhelp f this result; but we will prve mre: pp is the sle functin that cartperfrm the inversin in Therem 2. Fr this we have t prve the Then fllwing k M!M_4. If f(l) f 0 and E eln f (e)pjt (n/e ) fen), then p* (n) pen) fr every n E Z. Prf: See [2, lemma 4]. Suppse nw that ~~ is such that (7 ) g(n) = E fen/d), fen) E g(d)p.*(n/d) = E fee) l: ~*(6') ein d6 '=n 6e=d 6EQ writting p*(n) = E n =6d, p*(6) <=> fen) = l: g(d),..*(n/d). = E ~*(n/d)'l: fee) d ] n 6e=d 6EQ = E fee) E ~'It(6');. el n M '=n;e 6EQ we have fen) = f(e)p* (n/e), s by lemma 4 and therem 1, Thus we have prved the 10
!g~q~m_~.let f and such that g(l) I O. If g(n) = E f(n/d) <=> djn g be arithmetical functins f(n) = E g(d)~~(n/d) then 1"11" = tip REFERENCES 1. COHEN,E.: A class f residue systems (md r) and related arithmetical functins,i. A generalizatin Of(MObl)'USfunctin, f~21fi2_~~2f~m~~h., vl. 9 1959,13. 2. SATYANARAYANA, U.V.: On the inversin pr,perty f the MObiUS)~-functin, Th~_~~~h~Qe~~~~~' vl. XLVII(1963,38-42. ~~ ~E~~~~~~2_~~~~~~~~~~i2~~~Y~~~~~~~~~!2~ Universidad Nacinal de Clmbia trecibid~en-;n~;;-de~1968r-'---