ON THE FOURIER TRANSFORM OF THE GREATEST COMMON DIVISOR

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1 #A4 INTEGERS 13 (013) ON THE FOURIER TRANSFORM OF THE GREATEST COMMON DIVISOR Peter H. va er Kap Dept. of Matheatics a Statistics, La Trobe Uiversity, Victoria, Australia P.vaerap@latrobe.eu.au Receive: 5/3/1, Revise: 1/5/1, Accepte: 5/4/13, Publishe: 5/10/13 Abstract The iscrete Fourier trasfor of the greatest coo ivisor i[a]() gc(, )α a, with α a priitive -th root of uity, is a ultiplicative fuctio that geeralizes both the gc-su fuctio a Euler s totiet fuctio. O the oe ha it is the Dirichlet covolutio of the ietity with Raauja s su, i[a] i c (a), a o the other ha it ca be writte as a geeralize covolutio prouct, i[a] i a φ. We show that i[a]() couts the uber of eleets i the set of orere pairs (i, j) such that i j a o. Furtherore we geeralize a oze ow ietities for the totiet fuctio, to ietities which ivolve the iscrete Fourier trasfor of the greatest coo ivisor, icluig its partial sus, a its Labert series. 1. Itrouctio I [15] iscrete Fourier trasfors of fuctios of the greatest coo ivisor were stuie, i.e., h[a]() : h(gc(, ))α a, where α is a priitive -th root of uity. The ai result i that paper is the ietity 1 h[a] h c (a), where c (a) : gc(,)1 α a (1) 1 Siilar results i the cotext of r-eve fuctio were obtaie earlier; see [10] for etails.

2 INTEGERS: 13 (013) is Raauja s su, a eotes Dirichlet covolutio, i.e., h[a]() h c (a). () Raauja s su geeralizes both Euler s totiet fuctio φ c (0) a the Möbius fuctio µ c (1). Thus, ietity () geeralizes the forula h(gc(, )) (h φ)(), (3) alreay ow to Cesàro i The forula () shows that h[a] is ultiplicative if h is ultiplicative (because c (a) is ultiplicative a Dirichlet covolutio preserves ultiplicativity). Here we will tae h i to be the ietity fuctio (of the greatest coo ivisor) a stuy its Fourier trasfor. Obviously, as i() : is ultiplicative, the fuctio i[a] is ultiplicative, for all a. Two special cases are well-ow. Taig a 0 we have i[0] P, where P() : gc(, ) (4) is ow as Pillai s arithetical fuctio or the gc-su fuctio [4]. Secoly, by taig a 1 i (), we fi that i[1] i µ equals φ, by Möbius iversio of Euler s ietity φ u i, where u µ 1 is the uit fuctio efie by u() : 1. Let Fa eote the set of orere pairs of cogruece classes (i, j) such that i j a o, the set of factorizatios of a oulo. We clai that i[a]() couts its uber of eleets. Let us first cosier two special cases. a 0 For give i {1,,..., } the cogruece i j 0 o yiels i gc(i, ) j 0 o gc(i, ), which has a uique solutio oulo /gc(i, ), a so there are gc(i, ) solutios oulo. Hece, the total uber of eleets i F0 is P(). a 1 The totiet fuctio φ() couts the uber of ivertible cogruece classes oulo. As for every ivertible cogruece class i oulo there is a uique j i 1 o such that i j 1 o, it couts the uber of eleets i the set F 1. To prove the geeral case we eploy a Kluyver-lie forula for i[a], that is, a forula siilar to the forula for the Raauja su fuctio: c (a) µ. (5) gc(a,)

3 INTEGERS: 13 (013) 3 attribute to Kluyver [8]. Together the ietities () a (5) iply, cf. Sectio 3, i[a]() φ, (6) gc(a,) a we will show, i the ext sectio, that the uber #Fa of factorizatios of a o is give by the sae su. The right-ha sies of (5) a (6) are particular istaces of so-calle geeralize Raauja sus [1], a both forulas follow as cosequece of a geeral forula for the Fourier coefficiets of such sus [, 3]. I Sectio 3 we provie siple proofs for soe of the ice properties of these sus. I particular we iterpret the sus as a geeralizatio of Dirichlet covolutio. This iterpretatio lies at the heart of ay of the geeralize totiet ietities we establish i Sectio 4.. The Nuber of Factorizatios of a o We are after the uber of pairs (i, j) of cogruece classes oulo such that i j a o. (7) For give i, N, let g eote gc(i, ). If the cogruece i j a o has a solutio j, the g a a j i 1 a/g is uique o /g, so o there are g solutios. This yiels #Fa gc(i, ), which ca be writte as i1 gc(i,) a #F a a i1 gc(i,). (8) If the the su i1 gc(i,) 1 is epty. Now let. The oly itegers i which cotribute to the su are the ultiples of,, where gc(, /) 1. There are exactly φ(/) of the. Therefore the right-ha sies of forulae (6) a (8) agree, a hece #F a i[a]().

4 INTEGERS: 13 (013) 4 3. A Historical Rear a Geeralize Raauja Sus It is well-ow that Raauja was ot the first who cosiere the su c (a). Kluyver prove his forula (5) i 1906, twelve years before Raauja publishe the ovel iea of expressig arithetical fuctios i the for of a series s a sc s () [11]. It is ot well-ow that Kluyver actually showe that c (a) equals Vo Sterec s fuctio, itrouce i [13], i.e., µ φ() c (a) gc(a, ) φ gc(a, ). (9) This relatio is referre to i the literature as Höler s relatio, cf. the rear o page 13 i [1]. However, Höler publishe this relatio (9) thirty years after Kluyver [7]. We refer to [1, Theore ], or [, Theore 8.8] for a geeralizatio of (9). The so-calle geeralize Raauja sus, f a g() : gc(a,) f()g, (10) were itrouce i [1]. The otatio a is ew, the sus are eote S(a; ) i [1], s (a) i [], a S f,h (a, ) i [3]. I the cotext of r-eve fuctios [10] the sus are eote S f,g (a), a cosiere as sequeces of -eve fuctios, with arguet a. We cosier the sus as a sequece of fuctios with arguet, labele by a. We call f a g the a-covolutio of f a g. The cocept of a-covolutio provies a geeralizatio of Dirichlet covolutio as f 0 g f g. As we will see below shortly, the fuctio f a g is ultiplicative (for all a) if f a g are, a the followig iter-associative property hols, cf. [3, Theore 4]: (f a g) h f a (g h). (11) We also aopt the otatio f a : i a f, a call this the Kluyver, or a- extesio of f. Thus, we have f 0 i f, f 1 f, a forulas (5) a (6) becoe c (a) µ a (), a i[a] φ a, respectively. I ters of the Iverso bracet, with P a logical stateet, 1 if P, [P ] : 0 if ot P, the ietity fuctio I is give by I() : [ 1]. It is the ietity fuctio i the algebra where covolutio is the ultiplicatio, i.e., I f f I f. Let us cosier the fuctio f a I. It is f a I() f()[ ] [ a]f(). gc(a,)

5 INTEGERS: 13 (013) 5 Sice the fuctio [ a] is ultiplicative, the fuctio f a I is ultiplicative if f is ultiplicative. We ay write, cf. [3, eq. (9)], f a g() [ a]f()g (f a I) g(), (1) which shows that f a g is ultiplicative if f a g are. Usig equatio (1), the iter-associativity property (11) follows easily fro the associativity of Dirichlet covolutio. Proof. We have (f a g) h ((f a I) g) h (f a I) (g h) f a (g h). We ote that the a-covolutio prouct is either associative, or coutative. The iter-associativity a the coutativity of Dirichlet covolutio iply that f a g (f g) a f g a. (13) We provie a siple proof for forula (6), which states that the Fourier trasfor of the greatest coo ivisor is the Kluyver extesio of the totiet fuctio. Proof. Usig (), (5) a (13) we get i[a] i c (a) i µ a (i µ) a φ a. Forula (6) also follows as a special case of the forula for the Fourier coefficiets of a-covolutios, f a g() q[]()α a f, q[] : g i, (14) which was give i [1, ]. Forula (14) cobies with () a (5) to yiel a forula for fuctios of the greatest coo ivisor, h[] : h(gc(, )), aely h[] (h µ) u. (15) Proof. The Fourier coefficiets of h[a]() are h[](). But h[a] h c (a) (i a µ) h i a (µ h), a so, usig (14), the Fourier coefficiets are also give by (h µ) u(). For a Dirichlet covolutio with a Fourier trasfor of a fuctio of the greatest coo ivisor we have f g[a] f g[a]. (16) Proof. We have f g[a] f (g µ a ) (f g) µ a f g[a]. Siilarly, for a a-covolutio with a Fourier trasfor of a fuctio of the greatest coo ivisor, we have f a g[b] f a g[b]. (17) Proof. Iee, f a g[b] f a (g µ b ) (f a g) µ b f a g[b].

6 INTEGERS: 13 (013) 6 4. Geeralize Totiet Ietities The totiet fuctio is a iportat fuctio i uber theory, a relate fiels of atheatics. It is extesively stuie, coecte to ay other otios a fuctios, a there exist uerous geeralizatio a extesios, cf. the chapter The ay facets of Euler s totiet i [1]. The Kluyver extesio of the totiet fuctio is a very atural extesio, a it is ost surprisig it has ot bee stuie before. I this sectio we geeralize a oze ow ietities for the totiet fuctio φ, to ietities which ivolve its Kluyver extesio φ a, a..a. the iscrete Fourier trasfor of the greatest coo ivisor. This iclues a geeralizatio of Euler s ietity, the partial sus of φ a, a its Labert series The Value of φ a at Powers of Pries We start by proviig a forula for the value of φ a at powers of pries. This epes oly o the ultiplicity of the prie i a. The forulae, with p prie, P(p ) ( + 1)p p 1, φ(p ) p p 1, of which the first oe is Theore. i [4], geeralize to φ a (p ) (p p 1 )(l + 1) if l <, ( + 1)p p 1 if l, (18) where l is the largest iteger, or ifiity, such that p l a. Proof. We have which equals (18). φ a (p ) 4.. Partial Sus of φ a /i gc(p l,p ) i(l,) r0 To geeralize the totiet ietity φ p p r φ(p r ) l r0 p p 1 if l <, ( 1 r0 p p 1 ) + p if l, φ() µ(). (19)

7 INTEGERS: 13 (013) 7 to a ietity for φ a we first establish f 0 () f(). (0) Proof. Sice there are / ultiples of i the rage [1, ] it follows that f i() f() 1 f(). As a corollary we obtai Proof. Eployig (11) we fi f a (g i)() f a g 0 () f a g() (f a g) i(). (1) f a g(). Now taig f i a g µ i (1) we fi φ a () c (a). () 4.3. Partial Sus of P a /i Expresse i Ters of φ a Taig f i a g φ i (1) we fi P a () φ a (). (3) Note that by taig either a 0 i (), or a 1 i (3), we fi a ietity ivolvig the totiet fuctio a the gc-su fuctio, P() φ(). (4)

8 INTEGERS: 13 (013) Partial Sus of φ a To geeralize the totiet ietity, with > 0, φ() µ(), (5) we first establish f 0 () 1 f() + f u(). (6) Proof. By chagig variables, l, we fi (f i f u)() f()( 1) 1 / l1 f()(l 1) f(). 1 Note that this gives a ice proof of (5), taig f µ, as I() [ > 0]. Whe f µ a, the (13) iplies f i φ a, a f u I a, a therefore as a special case of (6) we obtai φ a () 1 [ ] + c (a). (7) a We rear that whe a we have a [ ] σ(a), where σ i u is the su of ivisors fuctio Geeralizatio of Euler s Ietity Euler s ietity, φ u i, geeralizes to φ a () τ(gc(a, )), (8) where τ u u is the uber of ivisors fuctio. Proof. We have φ a u (φ u) a i a, where i a () τ(gc(a, )). (9) gc(a,)

9 INTEGERS: 13 (013) Partial Sus of P a Expresse i Ters of φ a (a τ ) If f φ a, the f i P a, a (6) becoes, usig (8), P a () 1 τ(gc(a, )) Four Ietities Fro Césaro φ a (). (30) Accorig to Dicso [5] the followig three ietities were obtaie by Césaro: φ P(), (31) φ() 1 gc(j, ), (3) j1 φ()φ φ(gc(j, )). (33) Ietity (31), which is Theore.3 i [4], is obtaie by taig a 0 i (6), or h i i (3). It geeralizes to Proof. Tae f φ a g i i (13). j1 φ a Pa (). (34) Ietity (3) is obtaie by taig h 1/i i (3) a geeralizes to φ a() Proof. Tae f φ a g 1/i i (13). j1 gc(a,) 1 gc j,. (35) Ietity (33) is also a special case of (3), with h φ. It geeralizes to Proof. We have φ a ()φ b j1 gc(a,) φ b gc j,. (36) (i a φ) (i b φ) i a (i b (φ φ)) i a (i b φ[0]) i a φb [0],

10 INTEGERS: 13 (013) 10 a evaluatio at yiels gc(a,) / j1 φ b gc j, gc(a,) j1 φ b gc j,. The ore geeral ietity (3) geeralizes to h a (gc(, )) h φ a (). (37) 4.8. Three Ietities Fro Liouville Dicso [5, p.85-86] states, aogst ay others ietities that were presete by Liouville i the series [9], the followig ietities: φ()τ σ(), (38) φ()σ[ + 1] σ[](), (39) φ()τ θ, (40) where σ[] i[] u, i[]() :, a θ() is the uber of ecopositios of ito two relatively prie factors. All three are of the for φ f g a therefore they gai sigificace ue to (3), though Liouville ight ot have bee aware of this. For exaple, (3) a (38) cobie to yiel τ(gc(, )) σ(). The three ietities are easily prove by substitutig τ u u, σ[] i[] u, τ i[] θ u, φ µ i, a usig µ u I. They geeralize to φ a ()τ σa (), (41) φ a ()σ[ + 1] u a σ[](), (4) φ a ()τ τ(gc(a, ))θ. (43) These geeralizatios are prove usig the sae substitutios, together with (13), or for the latter ietity, (11) a (9). We ote that ee ot be iteger value i (39) a (4).

11 INTEGERS: 13 (013) Oe Ietity fro Dirichlet Dicso [5] writes that Dirichlet [6], by taig partial sus o both sies of Euler s ietity, obtaie + 1 φ(). By taig partial sus o both sies of equatio (8) we obtai φ a () a + 1 Proof. Suig the left-ha sie of (8) over yiels φ a () 1 1 φ a () a suig the right-ha sie of (8) over yiels τ(gc(a, )) 1 1 gc(a,) a / a. (44) + 1 / The Labert Series of φ a As show by Liouville [9], cf. [5, p.10], the Labert series of the totiet fuctio is give by x φ() 1 x x (1 x). 1 The Labert series for φ a is give by x φ a () 1 x p[a](x) x (1 x a ), (45) 1 where the coefficiets of p[a](x) a c[a]()x 1 are give by c[a] i a t[a], a t[a] is the piece-wise liear fuctio t[a]() a a. Proof. Cesàro prove x f() 1 x x f(), 1 1

12 INTEGERS: 13 (013) 1 cf. [5] a exercise 31 to chapter i [14]. By substitutig (8) i this forula we fi x φ a () 1 x x τ(gc(a, )). 1 1 Multiplyig the right-ha sie by (1 x a + x a ) yiels ( x τ(gc(a, ))) ( 1 + ( 1+a a+1 x τ(gc(a, ))( a)) x τ(gc(a, ))( a)) c[a]()x, where τ(gc(a, )) 0 < a, c[a]() τ(gc(a, ))( ( a)) τ(gc(a, ))(a ) a < a, τ(gc(a, ))( ( a) + a) 0 > a. Rewritig, usig (9), the fact that gc(a, a + ) gc(a, a ), a iviig by x, yiels the result. The polyoials p[a] see to be irreucible over Z a their zeros are i soe sese close to the a-th roots of uity; see Figures 4.10 a Figure 1: The roots of p[37] are epicte as crosses a the 37 th roots of uity as poits. This figure shows that whe a is prie the roots of p[a] that are close to 1 are closer to a th roots of uity.

13 INTEGERS: 13 (013) 13 Figure : The roots of p[35] are epicte as crosses a the 35 th roots of uity as poits. This figure shows that roots of p[a] are closest to priitive a th roots of uity A Series Relate to the Labert Series of φ a Liouville [9] also showe We show that 1 φ() 1 + x (1 + x x ) (1 x ). x x φ a () 1 + x q[a](x) x (1 x a ), (46) 1 where q[a](x) 4a b[a]()x 1, with b[a] h[a] t[a], a h[a]() i a () [ ]i a (/). Proof. As the left-ha sie of (46) is obtaie fro the left-ha sie of (45) by subtractig twice the sae series with x replace by x, the sae is true for the right-ha sie. Thus it follows that q[a](x) p[a](x)(1 + x a ) p[a](x )x, a hece, that b[a]() i a () [ ]i a (/) a, i a (a ) + i a ( a) [ ]i a (/) a < a, i a (3a ) + i a ( a) [ ]i a (a /) a < 3a, i a (4a ) [ ]i a (a /) 3a < 4a. The result follows ue to the ietities i a (a ) + i a ( a) i a (), i a (3a ) + i a ( a) i a (4a ), which are easily verifie usig (9).

14 INTEGERS: 13 (013) 14 We ca express the fuctios b[a] a h[a] i ters of a iterestig fractal fuctio. Let a fuctio κ of two variables be efie recursively by 0, a, or 0, κ[a]() κ[a/](/), a, (47) τ(gc(a, )). The followig properties are easily verifie usig the efiitio. We have a, with gc(a, b) 1, κ[a](a + ) κ[a](a ), (48) κ[a](b) 0 b, α() b, (49) where α eotes the uber of o ivisors fuctio, i.e., for all a o α( ) τ(). (50) Property (49) is a quite rearable fractal property; fro the origi i every irectio we see either the zero sequece, or α, at ifferet scales. We clai that h[a] κ[a]i (51) follows fro (47), (50), a (9). Fro (51) a (48) we obtai We ow prove that 1 + x ivies q[a] whe a is o. b[a] κ[a]t[a]. (5) Proof. Notig that b[a](a + ) b[a](a ) a, whe a, b[a]() 0, we therefore have q[a](x) a 1 b[a]( 1)x a b[a](a + 1)x a + b[a](a + 1)x a+ 1 a b[a](a + 1)x a (1 + x 4 ), 1 which vaishes at the poits where x 1. Apart fro the factor 1 + x whe a is o, the polyoial q[a] sees to be irreucible over Z a its zeros are i soe sese close to the (a) th roots of 1 or, to the (a + 1) st roots of uity, see Figure 4.11.

15 INTEGERS: 13 (013) 15 Figure 3: The roots of q[19] are epicte as boxes, the 38 th roots of 1 as poits, a the 0 th roots of uity as crosses A Perfect Square Our last ietity geeralizes the fait fact that φ(1) 1. We have φ a (). (53) a1 Proof. For ay lattice poit (i, j) i the square [1, ] [1, ] the prouct i j o is cogruet to soe a i the rage [1, ]. Acowleget. This research has bee fue by the Australia Research Coucil through the Cetre of Excellece for Matheatics a Statistics of Coplex Systes. Refereces [1] D.R. Aerso a T.M. Apostol, The evaluatio of Raauja s su a geeralizatios, Due Math. J. 0 (1953), [] T.M. Apostol, Itrouctio to Aalytic Nuber Theory, Spriger-Verlag (1976). [3] T.M. Apostol, Arithetical properties of geeralize Raauja sus, Pacific J. Math. 41 (197), [4] K.A. Brougha, The gc-su fuctio, Joural of Iteger Sequeces 4 (001), Art

16 INTEGERS: 13 (013) 16 [5] L.E. Dicso, History of the theory of ubers, Caregie Ist. (1919); reprite by Chelsea (195). [6] G.L. Dirichlet, Über ie Bestiug er ittlere Werte i er Zahletheorie, Abh. Aa. Wiss. Berli (1849), 78 81; also i Were, vol. (1897), [7] O. Höler, Zur Theorie er Kreisteilugsgleichug K (x) 0, Prace Mateatyczo Fizycze 43 (1936), [8] J.C. Kluyver, Soe forulae cocerig the itegers less tha a prie to, i: KNAW, Proceeigs 9 I, Astera (1906), [9] J. Liouville, Sur quelques séries et prouits ifiis, J. Math. Pures Appl. (1857), [10] L. Tóth a P. Hauae, The iscrete Fourier trasfor of r-eve fuctios, Acta Uiv. Sapietiae Math. 3 (011), 5 5. [11] S. Raauja, O Certai Trigooetric Sus a their Applicatios i the Theory of Nubers, Trasactios of the Cabrige Philosophical Society (1918), 59 76; also i: Collecte papers of Sriivasa Raauja, E. G.H. Hary et al, Chelsea Publ. Cop. (196), [1] J. Saor a B. Crstici, Haboo of Nuber Theory II, Kluwer Aca. Publ. (004). [13] R. Daublebsy Vo Sterec, Ei Aalogo zur aitive Zahletheorie, Sitzber, Aa. Wiss. Wie, Math. Naturw. Klasse, vol. I ll (Abt. IIa) (190), [14] H. S. Wilf, Geeratigfuctioology, Acaeic Press, eitio (1994). [15] W. Schra, The Fourier trasfor of fuctios of the greatest coo ivisor, Itegers 8 (008) A50.