Weighted Gcd-Sum Functions

Transcription

1 Joural of Iteger Sequeces, Vol. 14 (011), Article Weighte Gc-Sum Fuctios László Tóth 1 Departmet of Mathematics Uiversity of Pécs Ifjúság u Pécs Hugary a Istitute of Mathematics, Departmet of Itegrative Biology Uiversität für Boekultur Gregor Meel-Straße 33 A-1180 Wie Austria ltoth@gamma.ttk.pte.hu Abstract We ivestigate weighte gc-sum fuctios, icluig the alteratig gc-sum fuctio a those havig as weights the biomial coefficiets a values of the Gamma fuctio. We also cosier the alteratig lcm-sum fuctio. 1 Itrouctio The gc-sum fuctio, calle also Pillai s arithmetical fuctio (OEIS A018804) is efie by P() := gc(k,) ( N := {1,,...}). (1) The fuctio P is multiplicative a its arithmetical a aalytical properties are etermie by the represetatio P() = φ(/) ( N), () 1 The author gratefully ackowleges support from the Austria Sciece Fu (FWF) uer the project Nr. P0847-N18. 1

2 where φ is Euler s fuctio. See the survey paper [5]. Note that for every prime power p a (a N), P(p a ) = (a+1)p a ap a 1. (3) Now let P alter () := ( 1) k 1 gc(k,) ( N) (4) be the alteratig gc-sum fuctio. As far as we kow, the fuctio (4) was ot cosiere before. Furthermore, let ( ) P biom () := gc(k,) ( N) (5) k (OEIS A159068), where ( k) are the biomial coefficiets. Every term of the sum (5) is a multiple of, sice gc(k,) = kx + y with suitable itegers x,y a k ( ) ( k = 1 ) k 1 (1 k ). Note also the symmetry ( ) ( k gc(k,) = k) gc( k,) (1 k 1). More geerally, cosier the weighte gc-sum fuctios efie by P w () := w(k,)gc(k,) ( N), (6) where the weights are fuctios w : N C. I this paper we evaluate the alteratig gc-sum fuctio P alter (), euce a formula for the fuctio P biom () a ivestigate other special cases of (6). We also give a formula for the alteratig lcm-sum fuctio efie by L alter () := ( 1) k 1 lcm[k,] ( N). (7) Similar results ca be erive for the weighte versios of certai aalogs a geeralizatios of the gc-sum fuctio, see [5], but we cofie ourselves to the fuctio (6). Geeral results We first give the followig simple result. Propositio 1. i) Let w : N C be a arbitrary fuctio. The P w () = / φ() w(j,) ( N). (8) ii) Assume that there is a fuctio g : (0,1] C such that w(k,) = g(k/) (1 k ) a let G() = g(k/) ( N). The P w () = φ()g(/) ( N). (9)

3 Proof. i) Usig Gauss formula m = mφ() for m = gc(k,), groupig the terms of (6) a eotig k = j we obtai at oce P w () := ii) Use (8) a that w(k, ) = gc(k,)φ() / φ() w(j,). / / / w(j,) = g(j/) = g(j/(/)) = G(/). For w(k,) = 1 we reobtai formula formula (). I the ext sectio we ivestigate other special cases, icluig those alreay metioe i the Itrouctio. Remark. Cosier the fuctio R w () := gc(k,)=1 w(k,) ( N). (10) The, similar to the proof of i), ow with the Möbius µ fuctio istea of φ, R w () = w(k, ) = gc(k,)µ() / µ() w(j,). (11) If coitio ii) is satisfie, the we have R w () = µ()g(/) ( N). (1) We will also poit out some special cases of (11) a (1). 3 Special cases 3.1 Alteratig gc-sum fuctio Let w(k,) = ( 1) k 1 (k, N). The we have the fuctio P alter () efie by (4). Propositio 3. Let N a write = a m, where a N 0 := {0,1,,...} a m N is o. The {, if is o (a = 0); P alter () = a 1 ap(m) = a P(), if is eve (a 1). (13) a+ 3

4 Proof. Use formula (8). Here / / / S () := w(j,) = ( 1) j 1 = ( 1) j. If is o, the every ivisor of is also o a obtai S () = / ( 1)j = 1, where / is o. Hece, P alter () = φ() =. Now let be eve a let. For o, S () = / ( 1)j = 0, sice / is eve. For eve, S () = / 1 = /. We obtai that P alter () = eve φ() = where the first sum is P() (cf. ()), a the seco oe is m φ() a m = a P(m). φ() + o Usig (3), P() = P( a )P(m) = a 1 (a+)p(m) a euce φ(), P alter () = P()+ a P(m) = P(m)( a a 1 (a+)) = a 1 ap(m) = a a+ P(). Remark 4. More geerally, cosier the polyomial f (x) := gc(k,)x k 1, (14) i.e., put w(k,) = x k 1 (formally). The f (1) = P(), f ( 1) = P alter () a euce from Propositio 1, f (x) := (1 x ) φ()x 1 1 x. (15) 3. Logarithms as weights Let P log () := Propositio 5. For every N, (log k) gc(k, ). (16) P log () = P()log+ log(!/ )φ(/). (17) 4

5 Proof. Apply formula (8). For w(k,) = logk, / / w(j,) = log(j) = ( ) log+log!, hece P log () = ( ( ) φ() ) log+log!, a a short computatio leas to (17). Remark 6. Writig the expoetial form of (17), k gc(k,) = P() ( ) φ(/)!. (18) Compare this to the kow formula gc(k,)=1 k = φ() ( ) µ(/)!, (19) cf. [, p. 197, Ex. 4] (OEIS A001783). 3.3 Discrete Fourier trasform of the gc s Cosier w(k,) = exp(πikr/) (k, N), where r Z, a eote P (r) DFT () := exp(πikr/) gc(k, ), (0) represetig the iscrete Fourier trasform of the fuctio f(k) = gc(k,) (k N). Propositio 7. For every N, r Z, P (r) DFT () = gc(,r) φ(/). (1) Proof. Here exp(πikr/) = g(k/) with g(x) = exp(πirx). Usig formula (9) a that {, if r; exp(πirk/) = 0, otherwise; we obtai P (r) DFT () =,/ r φ() =, r φ(/). 5

6 Remark 8. Formula (1) ca be writte i the form P (r) DFT () = c / (r), () where c (k) are the Ramauja sums. Furthermore, () ca be extee for r-eve fuctios. See [4], [6, Prop. ]. Note that i the preset treatmet we o ot ee properties of the Ramauja sums a of r-eve fuctios. For r = 0 (more geerally, i case r) we reobtai from (1) formula (). For r = 1 we euce exp(πik/) gc(k, ) = φ() ( N), (3) which gives by writig the real a the imagiary parts, respectively, cos(πk/) gc(k, ) = φ() ( N), (4) si(πk/)gc(k,) = 0 ( N), (5) similar relatios beig vali for gc(,r) = 1. Formulae (3), (4), (5) were poite out i [4, Ex. 3]. 3.4 Biomial coefficiets as weights Let w(k,) = ( k) (k, N). The we have the fuctio Pbiom () efie by (5). Propositio 9. For every N, P biom () = φ() ( 1) l cos (lπ/). (6) Proof. Let ε j r = exp(πij/r) (1 j r) eote the r-th roots of uity. Usig the kow ietity /r k=0 ( ) = 1 kr r r (1+ε j r) = r cf. [1, p. 84], a applyig (8) we euce P biom () = / ( ) φ() = j l=1 r cos (jπ/r)cos(jπ/r) (,r N), (7) φ() ( ) cos (lπ/)cos(lπ/) 1 l=1 givig (6). = φ() l=1( 1) l cos (lπ/) φ(), 6

7 Note that (11) a (7) lea to the followig formula for the sequece OEIS A056188: R biom () := gc(k,)=1 ( ) = k µ() 3.5 Weights cocerig the Gamma fuctio Now let where Γ is the Gamma fuctio. P Gamma () := Propositio 10. For every N, P Gamma () = logπ Proof. This follows by (9) a by Γ logγ ( 1) l cos (lπ/) ( > 1). (8) l=1 ( ) k gc(k,), (9) (P() ) 1 log+ 1 φ() log. (30) ( ) k = (π) ( 1)/ 1/, ( N), which is a cosequece of Gauss multiplicatio formula. Remark 11. (30) ca be writte also as P Gamma () = logπ (P() ) 1 (φ log)(), (31) where eotes the Dirichlet covolutio. Note that φ log = µ i log = Λ i, where i() = ( N) a Λ is the vo Magolt fuctio. Writig the expoetial form, ( Γ ( )) gc(k,) k = (π) (P() )/ / φ()/. (3) Compare this to the followig formula give i [3]: gc(k,)=1 Γ ( ) k = (π)φ()/ exp(λ()/) = { (π) φ()/ / p, = p a (a prime power); (π) φ()/, otherwise. (33) 7

8 3.6 Further special cases It is possible to ivestigate other special cases, too. As examples we give the ext oes with weights regarig, amog others, the floor fuctio., a the saw-tooth fuctio ψ efie as ψ(x) = x x 1/ for x R\Z a ψ(x) = 0 for x Z. Propositio 1. For every N, P i () := Propositio 13. For every N a α R, P floor () := α+ k gc(k,) = Propositio 14. For every,r N, P (r) Propositio 15. For every N, > 1, kgc(k,) = (P()+). (34) α φ(). (35) saw-tooth () := ψ(kr/) gc(k, ) = 0. (36) 1 P si () := (logsi(kπ/))gc(k,) = (φ log)() (log)(p() ). (37) Propositio 16. For every N a α R with α+k/ / Z (1 k ), P cot () := cotπ(α+k/)gc(k,) = φ() cot(πα/). (38) These follow from Propositio 1 usig the followig well-kow formulae: α+ k = α, ( N), (39) ψ(kr/) = 0 (,r N), (40) 1 si(kπ/) = ( N) (41) 1 (for = 1 the empty prouct is 1), cotπ(α+k/) = cotπα ( N,α R,α+k/ / Z,1 k ). (4) 8

9 4 The alteratig lcm-sum fuctio Some of the previous results have couterparts for the lcm-sum fuctio (OEIS A051193) L() := lcm[k,] = 1+ φ() ( N). (43) We cosier here the alteratig lcm-sum fuctio efie by (7) a the the aalog of (18). Let F() := 1 φ(). Note that F() = (gc(k,)) 1 represetig the arithmetic mea of the orers of elemets i the cyclic group of orer, cf. [5, p. 3]. Furthermore, let β() := (1 µi)() = (1 p) a let h() := kk be the sequece of hyperfactorials (OEIS A00109). Propositio 17. Let N. If is o, the L alter () = 1+ µ()τ(/) = 1+ a (a(1 p)+1), (44) p where τ is the ivisor fuctio. If is eve of the form = a m, where a 1 a m N is o, the ( ) L alter () = a 1 a 1 m mf(m) 1 = ( ) a 1 3 a+1 +1 F() 1. (45) Proof. Let i 1 () = 1 a 1() = 1 ( N). We have L alter () = ( 1) k 1 1 k gc(k,) = ( 1) k 1 k = / β() ( 1) j 1 j. gc(k,) (i 1 µ)() Now usig that ( 1)k 1 k = ( 1) 1 ( + 1)/ ( N) the give formulae are obtaie alog the same lies with the proof of Propositio 3. Propositio 18. For every N, ( ) 1/ k lcm[k,] = h(/) β() β()/ / β()/ /. (46) 9

10 Proof. Similar to the proofs of above, = (logk)lcm[k,] = 1 (klogk) gc(k, ) / (klogk) 1 µ)() = gc(k,)(i (i 1 µ)() j log(j) equivalet to (46). Refereces = β()logh(/)+ β() log + 3 β() log, [1] L. Comtet, Avace Combiatorics. The Art of Fiite a Ifiite Expasios, D. Reiel Publishig Co., [] I. Nive, H. S. Zuckerma a H. L. Motgomery, A Itrouctio to the Theory of Numbers, 5th e., Joh Wiley & Sos, [3] J. Sáor a L. Tóth, A remark o the gamma fuctio, Elem. Math. 44 (1989), [4] W. Schramm, The Fourier trasform of fuctios of the greatest commo ivisor, Itegers 8 (008), #A50. [5] L. Tóth, A survey of gc-sum fuctios, J. Iteger Sequeces 13 (010), Article [6] L. Tóth a P. Haukkae, The iscrete Fourier trasform of r-eve fuctios, submitte, Mathematics Subject Classificatio: Primary 11A5; Secoary 05A10, 33B15. Keywors: gc-sum fuctio, lcm-sum fuctio, Euler s fuctio, Möbius fuctio, biomial coefficiet, Gamma fuctio. (Cocere with sequeces A001783, A00109, A018804, A051193, A056188, a A ) Receive May ; revise versio receive July Publishe i Joural of Iteger Sequeces, September Retur to Joural of Iteger Sequeces home page. 10