On the transcendence of infinite sums of values of rational functions

Transcription

1 O the trascedece of ifiite sus of values of ratioal fuctios N. Saradha ad R. Tijdea Abstract P () = We ivestigate coverget sus T = Q() ad U = P (X), Q(X) Q[X], ad Q(X) has oly siple ratioal roots. = ( ) P () Q() where Adhikari, Shorey ad the authors have show that T ad U are either ratioal or trascedetal. I the preset paper siple ecessary ad sufficiet coditios are forulated for the trascedece of T ad U if the degree of Q is 3 ad 2, respectively. Itroductio. Throughout the paper we take > to be a iteger ad f(x) a uber theoretic fuctio which is periodic od with f(i) = 0. We deote by Z, Q, Q, the rig of ratioal i= itegers, the field of ratioals ad the field of algebraic ubers. The first geeral result o the o-vaishig of S = = is probably due to Dirichlet who showed i 839 by his class uber forulas that L(, χ) = = χ() f() 0 if χ is a o-pricipal Dirichlet character. Aroud 970, Chowla ad Siegel obtaied soe o-vaishig results o periodic fuctios f assuig oly values, ad 0, see [4]. Chowla [4] ad Erdős (see [5]) forulated soe related cojectures, oe of which was proved by Baker, Birch ad Wirsig [3] i 973. They used Baker s theory o liear fors i logariths to establish that S 0 if f is a o-vaishig fuctio defied o the itegers with algebraic values ad period such that (i) f(r) = 0 if < gcd(r, ) <. (ii) the cyclotoic polyoial Φ is irreducible over Q(f(),..., f()).

2 If is prie, the (i) is vacuous ad if f is ratioal valued the (ii) holds trivially. Baker, Birch ad Wirsig further showed that their result would be false if (i) or (ii) is oitted. Ideed, let = p 2 where p is a prie ad f be defied by = f() = ( p s ) 2 ζ(s) where ζ(s) deotes the Riea zeta fuctio. For p = 2, this yields = 0 with period 4. Thus (i) is ecessary. To see that (ii) caot be oitted, they cosidered the uadratic characters χ, χ od 2 with coductors 3 ad 4, respectively, ad f = 2χ 3χ. The vaishig of S follows iediately fro the values L(, χ) = π 2 ad L(, 3 χ ) = π. They 3 also characterised all odd algebraic valued fuctios f for which S = 0. I 982 Okada [6] published a result which provides a descriptio of all fuctios for which (ii) holds ad S = 0. The criterio is a syste of ϕ() + ω() hoogeeous liear euatios i f(),..., f() with ratioal coefficiets where ϕ() deotes the Euler s totiet fuctio ad ω() is the uber of distict prie divisors of. The precise result is stated i Sectio 2. Okada s proof depeds o the basic result o the liear idepedece of the logariths of algebraic ubers ad o the o-vaishig of L(, χ) if χ is a o-pricipal Dirichlet character. Okada s result was used by Tijdea [7] to prove that S 0 if f : Z Q is copletely ultiplicative, ad if f is ultiplicative such that f(p k ) < p for every prie divisor p of ad every positive iteger k (see Sectio 2). I 200, Adhikari, Saradha, Shorey ad Tijdea [] proved that if S 0, the S is trascedetal. They used this result to prove that if P (X) Q[X] ad Q(X) Q[X] where Q(X) is a polyoial with siple ratioal roots, the T = = P () Q() is trascedetal provided T coverges ad is irratioal. They proved ay related results. Thus the uestio of decidig whether S = 0 or ot has gaied iportace. iforatio o the developets sketched above we refer to [2] ad [7]. For ore I the preset paper, we rephrase Okada s theore so that it becoes a decopositio f(a) lea (Lea ) ad use it to derive that S = 0 iplies = 0 for ay positive = iteger a coprie to (Corollary ). This is a iterediate result i [3, p.23, forula (7)]. 2

3 Lea ad Corollary are used i Sectio 3 to prove that ( ) (a + b) () ( + s )( + s 2 ) with a, b, s, s 2 Z, s s 2, a + b > 0 ad s, s 2 ever a o-egative iteger, is trascedetal except whe s s 2 (od ) ad a = 0. I the exceptioal case the su is ratioal. I Sectio 4, we use Lea to prove that a + b (2) ( + s )( + s 2 )( + s 3 ) with a, b, s, s 2, s 3 Z, s, s 2, s 3 distict, a 2 + b 2 0 ad s /, s 2 /, s 3 / ever a oegative iteger, is trascedetal except whe s s 2 s 3 (od ) or s s 2 (od ) ad as 3 = b or s s 3 (od ) ad as 2 = b or s 2 s 3 (od ) ad as = b. I the exceptioal cases the su is ratioal. O the other had, the exaples ( ) (3 + )(3 + 3)(3 + 5) = 6 ; (4 + )(4 + 2)(4 + 3)(4 + 4) = 0 show that the correspodig results for () ad (2) whe the deoiators are replaced with 3 ad 4 factors, respectively, are ot valid. 2 Decopositio Lea. We first itroduce soe otatio ad state Okada s ai result. We deote by P the set of all pries dividig. For p P ad Z, we deote by v p () the expoet to which p divides. We write J = {a Z a, gcd (a, ) = }, ad L = {r Z r, < gcd (r, ) < } L = L {}. We defie for r L ad p P, P (r) = {p P v p (r) v p ()} ad v p () + if p P (r) p ε(r, p) = v p (r) otherwise. 3

4 We further defie for r L ad a J, where ad Theore A. if A(r, a) = gcd(r, ) S(r) = p P (r) p P (r) ( p ϕ() ) S(r) p α(p) 0 α(p) < ϕ() if r a gcd(r, ) (od ) δ(r, a, ) = 0 otherwise δ(r, a, ) (Okada.) If Φ is irreducible over Q(f(),..., f()), the S = 0 if ad oly (3) f(a) + r L f(r)a(r, a) + f() ϕ() = 0 for a J ad (4) r L f(r)ε(r, p) = 0 for p P. We observe that (3) ad (4) for a syste of ϕ()+ω() hoogeeous liear euatios i f(),..., f() with o-egative ratioal coefficiets. Suppose f 0 ad S = 0. By (4), f() = 0 if f(r) = 0 for each r L. Hece, by (3), there exists at least oe r L for which f(r) 0. So Theore A iplies the result of Baker, Birch ad Wirsig etioed i Sectio. I particular we fid that if is prie, the S 0 i accordace with Chowla s cojecture. Now we give a euivalet versio of Theore A. Lea. (Decopositio lea.) Let Φ be irreducible over Q(f(),..., f()). Let M be the set of positive itegers which are coposed of prie factors of ad let v p () + ε(r, p) = if v p p(r) v p () v p (r) otherwise 4

5 The S = 0 if ad oly if (5) M f(a) = 0 for every a with 0 < a <, gcd(a, ) = ad (6) r= gcd(r,)> f(r)ε(r, p) = 0 for every prie divisor p of. Proof. show that Note that ε(r, p) 0 iplies p r, hece r L. So, by Theore A, it suffices to (7) f(a) + r L f(r)a(r, a) + f() ϕ() = f(a). M For ay iteger r, we deote by M(r) the set of positive itegers which are coposed of prie factors fro P (r). Thus M(r) = M(gcd(r, )) ad M = M(). We cosider f(a) f(a) f(a) = f(a) + M r L M a rod = f(a) + f(r) gcd(r, ) r L M(r) δ(r, a, ) + M + f() M() =: V. (8) We have M() = ( + p + p + ) = 2 p p p = ϕ(). Let M(r) ad δ(r, a, ) =. We have v p () = 0 if p P (r). If p P (r), the gcd(p, gcd(r,) ) = whece pϕ( gcd(r,) ) (od gcd(r,) ). Sice ϕ( gcd(r,) p ϕ() gcd(r, ) gcd(r, )(od ) whece δ(r, a, p ϕ() ) = δ(r, a, ). Thus (9) M(r) δ(r, a, ) = S(r) δ(r, a, ) p P (r) ( + p + + ). ϕ() p2ϕ() ) ϕ(), we obtai We substitute (8) ad (9) i the expressio V to obtai (7). [3]. As a coseuece of Lea we derive the followig corollary which is forula (7) of 5

6 Corollary. Let Φ be irreducible over Q(f(),, f()). Suppose S = 0. The = f(k) = 0 for every k with gcd(k, ) =. Proof. By Lea, we fid that (5) ad (6) hold. Thus, by (5), M f(ak) = M f(a ) = 0 for every a with a <, gcd(a, ) = where a ak (od ). If kr r (od ), the ε(r, p) = ε(r, p). Hece r L f(kr)ε(r, p) = r L f(r )ε(r, p) = 0 by (6). Now the assertio follows fro the coverse part of Lea. We write forally = f() = gcd(a,)= a M() f(a). It follows fro Lea that if the series o the left had side vaishes, so does the expressio withi brackets for every a coprie to. The coverse is ot true i geeral. For istace, if = 2 ad f() = ( ), Z, the for every odd a, but M(2) f(a) = f() f(2a) = f(a) + + f(4a) + = = ( ) = log 2 0. = I this exaple we have r L f(r)ε(r, p) = 2f(2) 0 ad hece (6) is ot satisfied. As aother coseuece of Lea, we derive Theores 9 ad 0 of [7]. Corollary 2. Let f be copletely ultiplicative, or ultiplicative with f(p k ) < p for every prie divisor p of ad every positive iteger k. Also let Φ be irreducible over Q(f(),..., f()). The S 0. 6

7 Proof. Suppose S = 0. Sice f is ultiplicative, (5) with a = iplies that M f() = 0. Note that the series is absolutely coverget because of the periodicity of f. Sice f() = ( ) f(p j ), p j M p j=0 we get (0). j=0 f(p j ) p j = 0 for soe prie divisor p of If f is copletely ultiplicative, the ( ) j f(p) = 0 for soe prie divisor p of j=0 p which is ot possible. If f is ultiplicative with f(p k ) < p for every prie divisor p of ad every positive iteger k, the j=0 f(p j ) p j > p p ( + p + ) = 0 cotradictig (0). We reark here that i the stateet of Theore 9 i [7], the coditio f(p k ) p should be replaced by f(p k ) < p. The exaple f(0) = 2, f() = f(5) =, f(2) = f(4) =, f(3) = 2 with period 6 shows that the stateet is false uder the forer coditio. The applicatio to Erdős proble i [7] is ot affected by this correctio. 3 Alteratig Series. I this sectio, we apply Lea ad Corollary to ivestigate sus () T = ( ) (α + β) ( + s )( + s 2 ) with α, β Q, s, s 2 Z. We prove 7

8 Theore. Suppose T is give by () with α + β > 0. Let Φ 2 be irreducible over Q(α, β) ad s, s 2 distict itegers such that +s, +s 2 do ot vaish for 0. Assue that α 0 if s s 2 (od ). The T is trascedetal. We derive Theore fro the followig lea. Lea 2. Let T = ( ) (α + β ) ( + r )( + r 2 ) with 0 < r, 0 < r 2, r r 2, r, r 2 Z, α, β Q. Suppose α + β > 0 ad Φ 2 is irreducible over Q(α, β ). The T is trascedetal. Proof. First we show that T 0. Suppose T = 0. We ay assue without loss of geerality that gcd(, r, r 2 ) =. By partial fractios, we see that T = { } A ( ) + B + r + r 2 where A = β α r (r r 2 ) ad B = β α r 2 (r r 2 ). Suppose A = 0. The B 0 ad T = B { }. 2 + r r 2 Hece T 0. Siilarly if A 0 ad B = 0, we have T 0. Thus we ay suppose that A 0, B 0. We defie f() for 0 as A if r (od 2) B if r 2 (od 2) f() = A if + r (od 2) B if + r 2 (od 2) 0 otherwise. Thus f is a periodic fuctio with period 2 ad (2) T = = f() = { f(r ) + f(r } 2) f(r ) f(r 2 ) = r 2 + r r r 2 Hece (5) ad (6) are valid by Lea. 8

9 Case. Let be odd. The oe of r i, + r i is eve ad the other is odd for i =, 2. Further ε(r i, 2) = 0 if r i is odd ad ε(r i, 2) = 2 if r i is eve. We apply (6) for p = 2 to obtai f(r ) = f(r 2 ). We re-write (2) as T = f(r ) { (2 + r )(2 + + r ) ± (2 + r 2 )(2 + + r 2 ) We see that all ters withi the curly brackets have the sae sig whece T 0, which is a cotradictio. Thus is ot odd. }. Case 2. Let be eve. The r i, + r i are both odd or both eve for i =, 2. By (5) ad (6), there exists a r L for which f(r) 0. Let r = r ad p be a prie factor of dividig r. The it follows fro (6) that there exists s 2 with p s ad s r such that f(s) 0. Now s r 2, s + r 2, sice gcd(, r, r 2 ) =. Hece s = + r. Assue gcd(r 2, 2) =. The gcd( +r 2, 2) =. Suppose v 2 (r ) v 2 (). The either v 2 (r ) > v 2 (), v 2 ( + r ) = v 2 () or v 2 (r ) = v 2 (), v 2 ( + r ) > v 2 (). O applyig (6) with p = 2, we obtai ε(r, 2) = ε( + r, 2). Sice v 2 (r ) v 2 ( + r ), this is possible oly whe v 2 (r ) v 2 (2) ad v 2 ( + r ) v 2 (2) which is false. Thus we get (3) v 2 (r ) = v 2 ( + r ) < v 2 (). Next we show that r 2, r 2 + p,, r 2 + ( p ) p are all coprie to 2. Suppose there exists a prie p dividig both r 2 + i p ad 2 for soe i with 0 < i < p. The p = p, v p () = ad there exists exactly oe such i, say i 0. Now we apply (5) with a = r 2 ad r 2 + i to see that p f(r 2 ) = f(r 2 + i p ) for 0 < i < p, i i 0, sice all the ters i (5) for > are eual because f(a) 0 for >, M oly whe a r or + r (od 2) ad p divides r ad. Sice r 2 ad + r 2 are the oly itegers k od 2 coprie to with f(k) 0, we coclude that p = 2. Thus r = R 2 α, + r = R 2 2 α with gcd(r, 2) = gcd(r 2, 2) = ad 0 < α < v 2 () by (3). We apply (5) with a = R, R 2 to get f(r ) + f(r 2 α ) 2 α = 0, f(r 2 ) + f(r 22 α ) 2 α = 0. 9

10 Hece f(r ) 0, f(r 2 ) 0. Thus {R, R 2 } = {r 2, + r 2 } which gives = R 2 R iplyig α = 0, which is a cotradictio. This proves that gcd(r i, 2) > for i =, 2. Note that r or r 2 is odd. Assue r is odd. (The case r 2 is odd is siilar.) Put d = gcd(r, 2) >. Hece d is odd. We put a = r d, b = +r d Now we show that it is possible to choose a iteger such that. The gcd(a, 2 (4) is prie, > 2, + (od 2), a b (od 2 d ). d ) = gcd(b, 2 d ) =. If a b + (od 2 ), the we choose ay prie > 2 which is d a b (od 2 ) ad (4) d is satisfied. So we suppose that a b + (od 2 2 ). There are ϕ(2)/ϕ( ) priitive d d residue classes od 2 i the residue class a b (od 2 ). Now d ϕ(2) ϕ( 2 ) d ( p ) 2 d p d p prie sice d is odd. Hece we ay take a priitive residue class + (od 2) ad a b (od 2 ). By Dirichlet s theore, we ca fid a prie satisfyig (4). Note that d (5) (r + ) = ad + bd + = r + + r (od 2). Sice gcd(, 2) =, we ay apply Corollary with k = to obtai { f(r ) (6) + f(r 2) f(r ) f(r } 2) = r 2 + r2 2 + r3 2 + r4 where r i r i (od 2), 0 < r i 2 for i 4 with r 3 = + r, r 4 = + r 2. We have (7) r 3 r 3 = ( + r ) ab bd = ad = r (od 2) ad by (5), (8) r r (( + r )) r 3 (od 2). Thus r 3 = r, r = r 3 = + r. Fro (5), it follows that ( )r 0 (od ). Suppose r 2 r 2 (od ). The ( )r 2 0 (od ). Hece (od ) sice gcd(, r, r 2 ) =. If (od 2), the a b(od 2 ) which is ot possible sice b a =. O the other d d had, + (od 2) by (4). Thus r2 r 2 (od ). Now we add (2) with (6) ad use (7), (8) to get { ( ) ( f(r 2 ) r r r2 0 ) } = r4

11 Note that r 4 r 4 = ( + r 2 ) + r 2 (od 2) so that r 4 = r 2 ±. By the ootoicity of with respect to k, the sig of the expressio betwee the 2+k 2+k+ curly brackets i the ifiite su above depeds oly o r 2 ad r2. Hece all these expressios have the sae sig. Sice r 2 r 2 (od ), the su does ot vaish. Hece f(r 2 ) = 0, which is a cotradictio. Thus T 0. Now we apply [, Theore ] to see that T is trascedetal. Proof of Theore. usig partial fractios, we have We assue without loss of geerality that gcd(, s, s 2 ) =. By T = { A ( ) + B } + s + s 2 where A = β αs (s s 2 ) ad B = β αs 2 (s s 2 ). Let s r (od ) ad s 2 r 2 (od ) with 0 < r, 0 < r 2. The (9) T = γ ± { A ( ) + B } + r + r 2 where γ Q(α, β). If r = r 2, the (20) T = γ ± α ( ) + r = γ ± α { }. 2 + r r We see that the ifiite su i the latter expressio for T does ot vaish. Now we apply [, Theore ] with if r (od 2) f() = if + r (od 2) 0 otherwise to coclude that the ifiite series ad hece T is trascedetal. Thus we ay assue that r r 2. Now we apply Lea 2 to (9) to see that T is trascedetal. We ote that if s s 2 (od ) ad α = 0, the by (20), we have T algebraic.

12 4 Series with ters havig three products i the deoiator. I this sectio we cosider series of the for α + β (2) U = ( + s )( + s 2 )( + s 3 ) with α, β Q, s, s 2, s 3 Z. We prove Theore 2. Suppose U is give by (2) with α + β > 0. Let Φ be irreducible over Q(α, β) ad s, s 2, s 3 be distict itegers such that + s, + s 2, + s 3 do ot vaish for 0. Assue that s, s 2, s 3 are ot i the sae residue class od. Further let (22) s s 2 (od ) if αs 3 = β; s 2 s 3 (od ) if αs = β; s 3 s (od ) if αs 2 = β. The U is trascedetal. We derive Theore 2 fro the followig lea. Lea 3. Let U = α + β ( + r )( + r 2 )( + r 3 ) with r, r 2, r 3 distict positive itegers ad α, β is irreducible over Q(α, β ). The U is trascedetal. Q. Suppose α + β > 0 ad Φ Proof. First we show that U 0. Suppose U = 0. We ay assue without loss of geerality that gcd(, r, r 2, r 3 ) =. Usig partial fractios, we get where A = U = { A β α r (r r 2 )(r r 3 ), B = + r + B + r 2 + C + r 3 β α r 2 (r 2 r )(r 2 r 3 ), C = } β α r 3 (r 3 r )(r 3 r 2 ). Observe that A + B + C = 0. Hece if ay of A, B, C vaishes the the series U reduces to soe series i which all the expressios betwee curly brackets have the sae sig. The U 0. Thus we ay assue that oe of A, B, C vaishes. We defie f() for 0 as 2

13 A if r (od ) B if r 2 (od ) f() = C if r 3 (od ) 0 otherwise. Thus f is a periodic fuctio with period takig oly three o-zero values f(r ), f(r 2 ), f(r 3 ) with (23) f(r ) + f(r 2 ) + f(r 3 ) = 0 ad U = = f() = 0. Hece (5) ad (6) are valid by Lea. Thus there exist r, s {r, r 2, r 3 } with r s ad a prie p dividig gcd(, r, s). Without loss of geerality, we ay take r = r, s = r 2. If p, by (6) with p = p, we have f(r ) + f(r 2 ) = 0 which gives f(r 3 ) = 0 by (23), which is a cotradictio. Thus p 2. Now suppose gcd(r 3, ) =. Let r 3 + i p a i (od ) where 0 < a i < for 0 i < p. We ote that a i s are all distict ad coprie to. Whe we apply (5) to the ubers a i, the all o-zero ters correspodig to > are the sae ad we fid f(a i ) = f(r 3 ) 0 for i = 0,,..., p. Sice p 2, this is a cotradictio. Thus gcd(r 3, ) >. Further we ay suppose that p r 3 sice gcd(, r, r 2, r 3 ) =. Let be a prie with gcd(r 3, ). By applyig (6) with p =, we ay assue that r 2. The r. Thus we have r = R p α, r 2 = R 2 p β γ, r 3 = R 3 δ where the dots represet other prie factors of ad gcd(r i, ) = for i =, 2, 3. Applyig (6) with p = p ad p =, we fid that f(r ) ad f(r 2) f(r 2 ) f(r 3 are egative ad, by (23), that ) f(r 2 ) = f(r ) + f(r 3 ). Hece f(r 2 ) > f(r ) ad f(r 2 ) > f(r 3 ). Agai usig (6) we get α > β, δ > γ ad v p (r 2 ) < v p (), v (r 2 ) < v (). It follows that gcd(r, r 3, ) = ad that v p (r 2 ) < v p () for every prie divisor p of. Hece we ay write r = R p α p αt t, r 2 = R 2 p β p βt t γ s γs, r 3 = R 3 δ s δs where p i s ad i s are distict sets of pries. By perutig r ad r 3 if ecessary, we ay assue that all the p i are odd. We select a uber a 0 coprie to as follows. Suppose R R 2 (od p β p βt t γ s γs 3 ).

14 The both the ubers R ± p β + are ot cogruet to R 2 (od the is coprie to. We take a 0 to be oe of R ± (od p β pβ t t γ γs s ivolvig f(r 2 ). Thus we get p β + p β pβ t t γ γs s ) ad oe of which is coprie to. If R R 2 ), the we take a 0 = R. Hece i (5) with a = a 0, o ter occurs 0 = Af(r ) + Bf(r 3 ) where A ad B represet certai o-vaishig series of positive ters. Thus f(r ) < 0 which f(r 3 ) is a cotradictio with f(r ) < 0, f(r 2) < 0. f(r 2 ) f(r 3 ) Proof of Theore 2. We assue without loss of geerality that gcd (, s, s 2, s 3 ) =. By usig partial fractios, we have where A = We observe that U = A { + + s B + s 2 + C + s 3 } β αs (s s 2 )(s s 3 ), B = β αs 2 (s 2 s )(s 2 s 3 ), C = β αs 3 (s 3 s )(s 3 s 2 ). (24) A + B + C = 0. Let s i r i (od ) with 0 < r i for i =, 2, 3. The we re-write (25) U = γ + A { + + r B + r 2 + C + r 3 } where γ Q(α, β). Suppose r = r 2. The r r 3 ad by (24), we have (26) U = γ + C(r r 3 ) ( + r )( + r 3 ). By our assuptio C 0. Hece the ifiite su i (26) does ot vaish. Now we apply [, Theore ] to coclude that U is trascedetal. Siilarly U is trascedetal wheever r = r 3 or r 2 = r 3. Thus we ay assue that r, r 2, r 3 are all distict. Now we apply Lea 3 to coclude that the ifiite su i (25) ad hece U is trasscedetal. We observe fro (22),(23),(24) that i the cases whe s, s 2, s 3 are all i the sae residue class (od ) or whe (22) is ot valid, the U is algebraic. 4

15 Refereces. [] S.D. Adhikari, N. Saradha, T.N. Shorey, R. Tijdea, Trascedetal ifiite sus, Idag. Math. N.S. 2 (200), -4. [2] S.D. Adhikari, Trascedetal Ifiite sus ad related uestios, to appear i Proc. Iter. Cof. o Nuber Theory at Chadigarh, October [3] A. Baker, B.J. Birch, E.A. Wirsig, O a proble of Chowla, J. Nuber Th. 5 (973), [4] S. Chowla, The Riea zeta ad allied fuctios, Bull. Aer. Math. Soc. 58(952), [5] A.E. Livigsto, The series f()/ for periodic f, Caad. Math. Bull. 8 (965), [6] T. Okada, O a certai ifiite sus for a periodic arithetical fuctios, Acta Arith. 40 (982), [7] R. Tijdea, Soe applicatios of diophatie approxiatios, to appear i Proc. Milleiu Cof. o Nuber Theory at Urbaa, May N. Saradha R. Tijdea School of Matheatics Matheatical Istitute Tata Istitute of Fudaetal Research Leide Uiversity Hoi Bhabha Road P.O. Box 952 Mubai RA Leide Idia The Netherlads eail: saradha@ath.tifr.res.i eail: tijdea@ath.leideuiv.l 5