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1 Ths ocument has been ownloae from Tampub The Insttutonal Repostory of Unversty of Tampere Publsher's verson Authors: Haukkanen Pentt Name of artcle: Some characterzatons of specally multplcatve functons Year of publcaton: 2003 Name of journal: Internatonal Journal of Mathematcs an Mathematcal Scences Volume: 2003 Number of ssue: 37 Pages: ISSN: Dscplne: Natural scences / Mathematcs Language: en School/Other Unt: School of Informaton Scences URL: URN: DOI: All materal supple va TamPub s protecte by copyrght an other ntellectual property rghts, an uplcaton or sale of all part of any of the repostory collectons s not permtte, except that materal may be uplcate by you for your research use or eucatonal purposes n electronc or prnt form. You must obtan permsson for any other use. Electronc or prnt copes may not be offere, whether for sale or otherwse to anyone who s not an authorze user.

2 IJMMS 2003:37, PII. S Hnaw Publshng Corp. SOME CHARACTERIZATIONS OF SPECIALLY MULTIPLICATIVE FUNCTIONS PENTTI HAUKKANEN Receve 10 January 2003 A multplcatve functon f s sa to be specally multplcatve f there s a completely multplcatve functon f A such that fm)fn)= m,n) fmn/ 2 )f A ) for all m an n. For example, the vsor functons an Ramanujan s τ-functon are specally multplcatve functons. Some characterzatons of specally multplcatve functons are gven n the lterature. In ths paper, we prove some further characterzatons of specally multplcatve functons Mathematcs Subject Classfcaton: 11A Introucton. An arthmetcal functon f s sa to be multplcatve f f1) = 1an fm)fn)= fmn) 1.1) whenever m,n) = 1. If 1.1) hols for all m an n, thenf s sa to be completely multplcatve. A multplcatve functon s known f the values fp n ) are known for all prme numbers p an postve ntegers n. A completely multplcatve functon s known f the values fp) are known for all prme numbers p. A multplcatve functon f s sa to be specally multplcatve f there s a completely multplcatve functon f A such that fm)fn)= m,n) ) mn f f A ) 1.2) 2 for all m an n, or equvalently fmn)= m,n) f ) ) m n f µ)f A ) 1.3) for all m an n, µ s the Möbus functon. If f A = δ, δ1) = 1an δn) = 0 for n>1,then 1.2) reuces to 1.1). Therefore, the class of completely multplcatve functons s a subclass of the class of specally multplcatve functons.

3 2336 PENTTI HAUKKANEN The stuy of specally multplcatve functons was ntate n [7], an arose n an effort to unerstan the entty σ α mn) = m,n) σ α m ) ) n σ α µ) α, 1.4) σ α n) enotes the sum of the αth powers of the postve vsors of n. Vayanathaswamy use the term quaratc functon, whle the term specally multplcatve functon was cone by Lehmer [3]. For more backgroun nformaton, reference s mae to the books by McCarthy [4] an Svaramakrshnan [6]. The Drchlet convoluton of two arthmetcal functons f an g s efne as f g)n) = nf)g ) n. 1.5) The functon δ serves as the entty uner the Drchlet convoluton. An arthmetcal functon f possesses a Drchlet nverse f 1 f an only f f1) 0. We next revew some basc characterzatons of specally multplcatve functons, see [4, 6]. Proposton 1.1. The followng statements are equvalent. 1) The functon f s a specally multplcatve functon. 2) The functon f s the Drchlet convoluton of two completely multplcatve functons a an b. In ths case f A = ab, the usual prouct of a an b.) 3) The functon f s a multplcatve functon, an for each prme number p, f 1 p n) = 0, n ) In ths case f A p) = f 1 p 2 ) for all prme numbers p.) 4) The functon f s a multplcatve functon, an for each prme number p, there exsts a complex number gp) such that f p n+1) = fp)f p n) gp)f p n 1), n ) In ths case f A p) = gp) for all prme numbers p.) 5) The functon f s a multplcatve functon, an for each prme number p, there exsts a complex number gp) such that f p n) n/2 ) n k [fp) = 1) k ] n 2k [ ] k, gp) n ) k k=0 In ths case f A p) = gp) for all prme numbers p.)

4 SOME CHARACTERIZATIONS OF SPECIALLY Remark 1.2. Completely multplcatve functons a an b npart 2 nee not be unque. The usual prouct ab, however, s unque. For example, let a, b, c, an be completely multplcatve functons such that ap) = 1anbp) = 2 for all prme numbers p, anc2) = 2, cp) = 1, 2) = 1, an p) = 2 for all prme numbers p 2. Then a b = c, but a,b c an a,b. However, ab = c. The purpose of ths paper s to prove some further characterzatons of specally multplcatve functons. As applcatons, we obtan formulas for the usual proucts σ α φ β, σ α σ β,anσ α τ, φ β s a generalze Euler totent functon an τ s Ramanujan s τ-functon. The functon φ β s gven by φ β = N β µ, N β n) = n β for all n. In partcular, we enote N 1 = N, N 0 = ζ, an φ 1 = φ, φ s the Euler totent functon. Ramanujan s τ-functon s a specally multplcatve functon wth τ A = N 11. In the characterzatons, we nee the concepts of the untary convoluton an the kth convolute. The untary convoluton of two arthmetcal functons f an g s efne as f g)n) = nf)g ) n, 1.9) n means that n,,n/) = 1. The kth convolute of an arthmetcal functon f s efne as Ω k f )n) = fn 1/k ) f n s a kth power, an Ω k f )n) = 0 otherwse. 2. Characterzatons Theorem 2.1. If f s a specally multplcatve functonang s a completely multplcatve functon, then h fg µ) = fg, 2.1) h s the specally multplcatve functon such that hp) = fp), h A p) = gp)f A p) 2.2) for all prme numbers p. Conversely, f f1) = 1 an there exst completely multplcatve functons a, b, g, an k such that a b fg µ) = fg, 2.3) ap)+bp) = f p), ap)bp) = gp)kp), g µ)n) gn) 2.4) for all prme numbers p an ntegers n 2), then f s a specally multplcatve functon wth f A = k.

5 2338 PENTTI HAUKKANEN Proof. By multplcatvty, t suffces to show that 2.1) hols at prme powers, that s, [ fg µ) ] p e ) = fg h 1) p e) 2.5) for all prme powers p e.ife = 1, then both ses of 2.5) are equal to fp)gp) fp). Assume that e 2. Then fg h 1 ) p e) = f p e) g p e) +f p e 1) g p e 1) h 1 p) +f p e 2) g p e 2) h 1 p 2) = f p e) g p e) f p e 1) g p e 1) fp) +f p e 2) g p e 2) gp)f A p). 2.6) By 1.7), we obtan fg h 1 ) p e) = f p e) g p e) f p e) g p e 1) = f p e) g µ) p e). 2.7) Thus we have prove 2.5). To prove the converse, we wrte 2.3) n the form fg µ) ) n) = fg a 1 b 1) n). 2.8) We wrte n = p e+1 e 1) an, after some smplfcatons, obtan f p e+1) = f p e) fp) f p e 1) kp). 2.9) Therefore, by 1.7), t remans to prove that f s multplcatve. Denote n = p e 1 1 per r p r +1 p r +s, e > 1 = 1,2,...,r). We procee by nucton on e 1 + +e r +s to prove that fn)= f p e 1 1 ) ) ) f p e r r f pr +1 ) f pr +s. 2.10) If e 1 + +e r + s = 1, then 2.10) hols. Suppose that 2.10) hols when e 1 + +e r +s<m.thenfore 1 + +e r +s = m, we have after some manpulaton f n)g µ)n) = fg a 1 b 1) n) n f = f n)gn)+ n >1 = f n)gn) p e n ) g ) n a 1 b 1) ) f p e) g p e) + fg a 1 b 1) p e) p e n

6 SOME CHARACTERIZATIONS OF SPECIALLY = f n)gn) f p e) g p e) + r =1 [ f p e p e n ) g p e ) f p e 1 )f p ) g s ) ) )) f pr + g pr + f pr +. =1 Usng 2.9), we obtan p e 1 ) +f p e 2 )k p ) g f n)g µ)n) = f n)gn) gn) f p e) +g µ)n) f p e). Ths gves 2.10). p e n p e n p e 1 )] 2.11) 2.12) Remark 2.2. The converse part of Theorem 2.1 can also be wrtten as follows. If f1) = 1 an there exst completely multplcatve functons g an k, an a specally multplcatve functon h such that h fg µ) = fg, 2.13) hp) = fp), h A p) = gp)kp), g µ)n) gn) 2.14) for all prme numbers p an ntegers n 2), then f s a specally multplcatve functon wth f A = k. Corollary 2.3. If f s a specally multplcatve functon, then h fφ= fn, 2.15) h s the specally multplcatve functon such that hp) = fp), h A p) = pf A p) 2.16) for all prme numbers p. Conversely, f f1) = 1 an f there exst completely multplcatve functons a, b, an k such that a b fφ= fn, 2.17) ap)+bp) = f p), ap)bp) = pkp) 2.18) for all prme numbers p, then f s a specally multplcatve functon wth f A = k.

7 2340 PENTTI HAUKKANEN Corollary 2.4. If f an g are completely multplcatve functons, then f fg µ) = fg. 2.19) Conversely, f f1) = 1 an f there exsts a completely multplcatve functon g such that f fg µ) = fg, 2.20) g µ)n) gn) 2.21) for all ntegers n 2), then f s a completely multplcatve functon. Corollary 2.5 Svaramakrshnan [5]). multplcatve functon f an only f If f1) = 1, then f s a completely f fφ= fn. 2.22) Example 2.6. We have σ α φ β = σ α N β h 1, 2.23) h s the specally multplcatve functon such that for all prme numbers p. hp) = σ α p) = p α +1, h A p) = p β p α = p α+β 2.24) Theorem 2.7. If f s a specally multplcatvefuncton ang s a completely multplcatve functon, then fg µ) = fg µf Ω 2 µ 2 f A g )). 2.25) Conversely, f f1) 0 an f there exst completely multplcatve functons g an k such that fg µ) = fg µf Ω 2 µ 2 kg )), 2.26) g µ)n) gn) 2.27) for all n, then f s a specally multplcatve functon wth f A = k. Proof. We observe that µf Ω2 µ 2 f A g )) p) = fp), µf Ω2 µ 2 f A g )) p 2) = f A p)gp), µf Ω2 µ 2 f A g )) p n) = )

8 SOME CHARACTERIZATIONS OF SPECIALLY for all prme numbers p an ntegers n 3). Therefore µf Ω 2 µ 2 f A g) = h 1, h s the specally multplcatve functon n Theorem 2.1. Thus 2.25) follows from 2.1). The converse follows from Theorem 2.1 snce µf Ω 2 µ 2 gk) = a 1 b 1, a an b are completely multplcatve functons as gven n Theorem 2.1. Theorem 2.8. If f s a specally multplcatve functonang s a completely multplcatve functon, then fg µ) = fg f 1 Ω 2 µ 2 f A g µ) )). 2.29) Conversely, f f1) = 1 an there exst completely multplcatve functons c,, an g such that fg µ) = fg c ) 1 Ω 2 µ 2 cg µ) )), 2.30) cp)+p) = fp), g µ)n) gn) 2.31) for all prme numbers p an ntegers n 2), then f s the specally multplcatve functon gven as f = c. Proof. Proof of Theorem 2.8 s smlar to that of Theorem 2.7. Example 2.9. We have σ α φ β = σ α N β µσ α Ω 2 µ 2 N α+β)), σ α φ β = σ α N β σα 1 Ω 2 µ 2 N α N β µ ))). 2.32) Lemma Suppose that f s an arthmetcal functon such that f1) = 1 an f 1 p ) = 0 for 3 <kk 4). Then f p k) = fp)f p k 1) f 1 p 2) f p k 2) f 1 p k). 2.33) Proof. Lemma 2.10 follows from the equaton k f 1 p ) f p k ) = ) =0 Theorem If f s a specally multplcatve functon an g s a completely multplcatve functon, then fg ζ) = fg f Ω 2 fa g ) )

9 2342 PENTTI HAUKKANEN Conversely, f f s a multplcatve functon such that fg ζ) = fg f Ω 2 hg) 1, 2.36) g s a completely multplcatve functon wth gp)g ζ)p e ) 0 for all prme powers p e an h s a completely multplcatve functon, then f s a specally multplcatve functon wth f A = h. Proof. Let f = a b, a an b are completely multplcatve functons. It s known [7] that fg ζ) = a b)g ζ) = ag aζ bg bζ Ω 2 abgζ) ) Usng elementary propertes of arthmetcal functons, we obtan fg ζ) = a b)g a b) Ω 2 fa g ) 1 = fg f Ω2 fa g ) ) Ths proves 2.35). Assume that 2.36) hols. Then 2.36) atp 2 gves hp) = fp) 2 f p 2). 2.39) Snce f 1 p 2 ) = fp) 2 fp 2 ) for all multplcatve functons, we obtan hp) = f 1 p 2). 2.40) We next prove that f 1 p ) = ) We procee by nucton on. Calculatng 2.36) atp 3 an usng 2.40) gves f p 3) = fp)f p 2) fp)f 1 p 2). 2.42) Snce f p 3) f p 2) fp)+fp)f 1 p 2) +f 1 p 3) = 0, 2.43) we see that f 1 p 3 ) = 0. Suppose that f 1 p ) = 0 for all 3 <kk>3). We wrte 2.36) as fg ζ) f 1 = fg Ω 2 hg) )

10 SOME CHARACTERIZATIONS OF SPECIALLY Suppose that k s even, say k = 2e e >1). At p 2e, the left-han se of 2.44) becomes 2e f p ) g ζ) p ) f 1 p 2e ) =0 = f 1 p 2e) +f p 2e 2) g ζ) p 2e 2) f 1 p 2) +f p 2e 1) g ζ) p 2e 1) f 1 p)+f p 2e) g ζ) p 2e) = f 1 p 2e) f 1 p 2e) g ζ) p 2e 2) fp)f p 2e 1) g p 2e 1) +f p 2e) g p 2e 1) +f p 2e) g p 2e) = f 1 p 2e) f 1 p 2e) g ζ) p 2e 1) 2.45) f 1 p 2) f p 2e 2) g p 2e 1) +f p 2e) g p 2e), the last two equatons are erve by Lemma Further, at p 2e,the rght-han se of 2.44) becomes 2e f p 2e ) g p 2e ) Ω 2 hg) 1 p ) =0 e = f p 2e )) g p 2e )) µ p ) h p ) g p ) =0 2.46) = f p 2e) g p 2e) f p 2e 1)) g p 2e 1)) hp)gp). Now, we see that f 1 p 2e ) = 0, that s, f 1 p k ) = 0. If k s o, a smlar argument apples. Thus 2.41) hols an therefore, by 1.6), f s a specally multplcatve functon wth f A = h. Corollary If f s a specally multplcatve functon, then fσ 0 = f f Ω 2 fa ) ) Conversely, f f s a multplcatve functon such that fσ 0 = f f Ω 2 h) 1, 2.48) h s a completely multplcatve functon, then f s a specally multplcatve functon wth f A = h. Corollary 2.13 Apostol [1]). functons, then If f an g are completely multplcatve fg ζ) = fg f. 2.49) Conversely, f f s a multplcatve functon such that fg ζ) = fg f, 2.50)

11 2344 PENTTI HAUKKANEN g s a completely multplcatve functon wth gp)g ζ)p e ) 0 for all prme powers p e, then f s a completely multplcatve functon. Corollary 2.14 Carltz [2]). Suppose that f s a multplcatve functon. Then f s a completely multplcatve functon f an only f fσ 0 = f f. 2.51) Corollary There exst τσ α = τn α τ Ω 2 N α+11 ) 1, σ α σ β = σ α N β σ α Ω 2 N α+β ) 2.52) 1. References [1] T. M. Apostol, Some propertes of completely multplcatve arthmetcal functons, Amer. Math. Monthly ), [2] L. Carltz, Completely multplcatve functon, Amer. Math. Monthly ), [3] D. H. Lehmer, Some functons of Ramanujan, Math. Stuent ), [4] P. J. McCarthy, Introucton to Arthmetcal Functons, Unverstext, Sprnger- Verlag, New York, [5] R. Svaramakrshnan, Multplcatve functon an ts Drchlet nverse, Amer. Math. Monthly ), [6], Classcal Theory of Arthmetc Functons, Monographs an Textbooks n Pure an Apple Mathematcs, vol. 126, Marcel Dekker, New York, [7] R. Vayanathaswamy, The theory of multplcatve arthmetc functons, Trans. Amer. Math. Soc ), no. 2, Pentt Haukkanen: Department of Mathematcs, Statstcs an Phlosophy, Unversty of Tampere, FIN-33014, Fnlan E-mal aress: Pentt.Haukkanen@uta.f