We establish a new multiplicity lemma for solutions of a differential system
extending Ramanujan's classical differential relations. This result can be
useful in the study of arithmetic properties of values of Riemann zeta function
at odd positive integers (Nesterenko, 2011)

Zorin, Evgeniy

English

arXiv

Numérisation de Documents Anciens Mathématiques

Multiplicity estimate for solutions of extended Ramanujan’s system

Evgeniy Zorin

Journal de Théorie des Nombres de Bordeaux

Evgeniy ZORINMultiplicity estimate for solutions of extended Ramanujan’s systemTome 24, no 3 (2012), p. 773-781.<http://jtnb.cedram.org/item?id=JTNB_2012__24_3_773_0>© Société Arithmétique de Bordeaux, 2012, tous droits réservés.L’accès aux articles de la revue « Journal de Théorie des Nom-bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accordavec les conditions générales d’utilisation (http://jtnb.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sousquelque forme que ce soit pour tout usage autre que l’utilisation àfin strictement personnelle du copiste est constitutive d’une infrac-tion pénale. Toute copie ou impression de ce fichier doit contenir laprésente mention de copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Journal de Théorie des Nombresde Bordeaux 24 (2012), 773-781Multiplicity estimate for solutions of extendedRamanujan’s systempar Evgeniy ZORINRésumé. Nous établissons un nouveau lemme de multiplicitépour les solutions d’un système différentiel généralisant les rela-tions différentielles classiques de Ramanujan. Ce résultat peut êtreutile pour l’étude des propriétés arithmétiques des valeurs de lafonction zêta de Riemann aux entiers positifs impairs (Nesterenko,2011).Abstract. We establish a new multiplicity lemma for solutionsof a differential system extending Ramanujan’s classical differen-tial relations. This result can be useful in the study of arithmeticproperties of values of Riemann zeta function at odd positive in-tegers (Nesterenko, 2011).1. IntroductionIn what follows we denote by σk(n), k ∈ Z, n ∈ N the sum of kth powersof divisors of n:σk(n) :=∑d|ndk.In this paper we consider the following sets of functions. First of all, Fourierseries of the Eisenstein functions(1.1) E2k(z) := 1−4kB2k∞∑n=1σ2k−1(n)zn, k ∈ N,where B2k are Bernoulli numbers. Also we consider(1.2) gu,v(z) :=∞∑n=1nuσ−v(n)zn, u, v ∈ N, 0 ≤ u < v, v is odd.We recall the following well-known fact: functions E2, E4 and E6 arealgebraically independent over C(z) and all the other functions E2k, k ≥ 4belong to Q[E4, E6] (see for instance [8]). More precisely, for all k ≥ 4 thereexists a polynomial Ak ∈ C[X,Y ] such thatE2k(z) = Ak(E4(z), E6(z))Manuscrit reçu le 13 septembre 2011.Classification math. 11J81, 11J82, 11J61.774 Evgeniy Zorinand these polynomials Ak(X,Y ), k ≥ 4 contain only monomials M of bi-degrees (degXM, degY M) satisfying 2 degXM + 3 degY M = k.In [2] Yu.Nesterenko proved the following theorem:Theorem 1.1 (Nesterenko). For any complex q, 0 < |q| < 1, the setq, E2(q), E4(q), E6(q)contains at least three algebraically independent over Q numbers.In particular, this result allowed him to establish the algebraic indepen-dence of π, eπ and Γ(14). Indeed, using modular properties of Eisensteinseries one can easily calculateE2(e−2π) =3π, E4(e−2π) = 3Γ(14)8(2π)6 , E6(e−2π) = 0(see Chapter 1 of [4]). Hence applying Theorem 1.1 with q = e−2π oneobtains algebraic independence of π, eπ and Γ(14).Later Yu.Nesterenko [3] showed that considering a bigger set of functions,notably introducing functions (1.2), we can capture other extremely inter-esting cases, for instance one has ζ(4n + 3) ∈ Q(E2(e−2π), g0,4n+3(e−2π))(n ∈ N). So extending Theorem 1.1 for a set of functions likeE2, E4, E6, gu,v, 0 ≤ u < v, v ≡ 3 mod 4one could potentially illuminate the question of transcendence or irrational-ity of ζ at odd positive points. See [3] for some more discussion.An important stage in the proof of Theorem 1.1 is the following multi-plicity lemma:Theorem 1.2 (Nesterenko). For any polynomial A(z,X1, X2, X3) ∈C[z,X1, X2, X3], A 6= 0, the inequality holdsordz=0A(z, P (z), Q(z), R(z)) < c(degz A+ 1)(degX A+ 1)3,where c is an absolute constant.We invite the interested reader to consult Chapters 3 and 10 in [4] aswell as [6, 7, 9] concerning the use of multiplicity estimates in algebraicindependence proofs.In this paper we adopt the method from [2] and [4, Chapter 10] to es-tablish (for any fixed odd m ≥ 3) a multiplicity lemma for the whole set offunctions(1.3) E2(z), E4(z), E6(z), gu,v(z), v is odd, 0 ≤ u < v ≤ m,see Theorem 2.1 below. Note that algebraic independence over C(z) of theset of functions (1.3) was already established in 2010 by P. Kozlov ([1],Multiplicity estimate 775for a statement see also [3], page 2), we use this result in the proof ofTheorem 2.1.Proofs of Theorems 1.1 and 1.2 are based on the fact that functions E2,E4, E6 satisfy the following system of differential equations:(1.4)δE2 =112(E22 − E4), δE4 =13 (E2E4 − E6) , δE6 =12(E2E6 − E24),where δ := z ddz .The set of functions (1.3) satisfy the following extended system of differ-ential equations [3]. We keep the three equations (1.4) and for any odd v ≥ 3(1.5)δgu,v(z) = gu+1,v(z), 0 ≤ u < v − 1,δgv−1,v(z) = Bv+1Av+1(E4(z), E6(z))− 12v + 2 ;in the case v = 1 we have the equation(1.6) δg0,1(z) =124 (1− E2(z)) .2. Multiplicity LemmaLet m ∈ N be a fixed positive odd integer. We introduce the followingnotation:(2.1) R := C[z,X1, X2, X3, Y0,1, Y0,3, Y1,3, Y2,3, . . . , Ym−1,m].Theorem 2.1. Let m ≥ 1 be an odd integer. There exists a constant Cdepending on m only such that for all non-zero P ∈ Rordz=0P (z, E2(z), E4(z), E6(z),g0,1(z), g0,3(z), . . . , g0,m(z), . . . , gm−1,m(z))≤ C (degz P + 1)(degX,Y P + 1)( m+12 )2+3 ,(2.2)where degX,Y P denotes the total degree of P in the variables X1, X2, X3,Y0,1, . . . , Ym−1,m, i.e. all the variables appearing in the definition (2.1) ofR but z.Remark 2.1. The exponent(m+12)2+ 3 in the r.h.s. of (2.2) equals thenumber of functions different than z in the l.h.s. of (2.2) (and also thetranscendence degree of R over C(z)). It is an easy exercise from linearalgebra that for any given set f1, . . . , fl of functions analytic at z = 0 andany N ∈ N, N ≥ 1, one can construct a polynomial PN,f ∈ C[z,X1, . . . , Xl]of degree degP ≥ N and satisfyingordz=0PN,f (z, f1(z), . . . , fl(z)) ≥ C0(degz PN,f + 1) (degX PN,f + 1)l,where C0 = 1(l+1)! .776 Evgeniy ZorinHence Theorem 2.1 provides multiplicity estimate for the set of func-tions (1.3) with the optimal exponent.In the sequel we denoteD0 := zddz+ 112(X21 −X2) ddX1+ 13 (X1X2 −X3)ddX2+ 12(X1X3 −X22) ddX3,D1 :=124 (1−X1)ddY0,1,Dv :=v−2∑k=0Yk+1,vddYk,v+Bv+1Av+1(X2, X3)− 12v + 2ddYv−1,v,v = 3, 5, . . . ,m, andD := D0 +(m−1)/2∑k=0D2k+1.(2.3)The differential operator D satisfies(2.4) DP (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))= z ddzP (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)).We deduce Theorem 2.1 using Nesterenko’s conditional Multiplicity Lem-ma (see Theorem 2.2 below). This result deals with differential system ofthe following type:(2.5) f ′i(z) =Ai(z, f)A0(z, f), i = 1, . . . , n,where Ai(z,X1, . . . , Xn) ∈ C[z,X1, . . . , Xn] for i = 0, ..., n (we suppose thatA0 is a non-zero polynomial).Remark 2.2. It is easy to see that the system (1.4)∪(1.5)∪(1.6) is of thetype (2.5).One associates to the system (2.5) the differential operator(2.6) DA = A0(z,X1, . . . , Xn)∂∂z+n∑i=1Ai(z,X1, . . . , Xn)∂∂Xi.In our case (i.e. the case of the system (2.5)) this formula gives exactly thedifferential operator D as defined in (2.3).Multiplicity estimate 777Theorem 2.2 (Nesterenko, see Theorem 1.1, Chapter 10 [4]). Suppose thatfunctionsf = (f1(z), . . . , fn(z)) ∈ C[[z]]nare analytic at the point z = 0 and form a solution of the system (2.5).If there exists a constant K0 such that every DA-stable prime ideal P ⊂C[X ′1, X1, . . . , Xn], P 6= (0), satisfies(2.7) minP∈Pordz=0P (z, f) ≤ K0,then there exists a constant K1 > 0 such that for any polynomial P ∈C[X ′1, X1, . . . , Xn], P 6= 0, the following inequality holds(2.8) ordz=0(P (z, f)) ≤ K1(degX′ P + 1)(degX P + 1)n.To deduce Theorem 2.1 from Theorem 2.2 it is sufficient to prove Propo-sition 2.1 here below.Proposition 2.1. If P is a non-zero prime ideal ofR = C[z,X1, X2, X3, Y0,1, . . . , Ym−1,m]with DP ⊂ P, then either z ∈ P or ∆ = X32 −X23 ∈ P.Proof of Theorem 2.1 modulo Proposition 2.1. If we have the result an-nounced in Proposition 2.1, then any prime D-stable ideal P contains thepolynomial(2.9) Θ := z∆ = z(X32 −X23).In this case we have obviously(2.10) minP∈Pordz=0P (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))≤ ordz=0Θ(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) = 2,as Θ(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) = z∆(E2(z), E4(z)) bydefinition (2.9) and ordz=0∆(E2(z), E4(z)) = 1. The upper bound (2.10)assures the hypothesis (2.7) of Theorem 2.2, and applying this theorem weobtain the upper bound (2.8), that is (2.2) in our case. To prove Proposition 2.1, we describe at first principal D-stable idealsof R.Lemma 2.1. There exist only two non-zero D-invariant principal primeideals of R, namely, the ideals generated by z and ∆.Proof. Suppose that A ∈ R is any irreducible polynomial with the propertythat A|DA. Thus(2.11) DA = AB, B ∈ R.778 Evgeniy ZorinWe readily verify with the definition of D (see (2.3)) that degY DA ≤degY A and degzDA ≤ degz A, hence (2.11) implies degY B = degz B = 0,i.e. B ∈ C[X1, X2, X3].For any F ∈ R we define the weight of F as φ : R→ N,φ(F ) := degt F (z, tX1, t2X2, t3X3, tm+1Y ).Then φ satisfies the following properties:(1) For any F ∈ Rφ(DF ) ≤ φ(F ) + 1.(2) For any F,G ∈ Rφ(FG) = φ(F ) + φ(G).These properties together with (2.11) implyφ(A) + φ(B) = φ(DA) ≤ φ(A) + 1,hence φ(B) ≤ 1. Thus B ∈ C[X1] and degB ≤ 1, i.e. B = aX1 +b, a, b ∈ C.So (2.11) in fact means(2.12) DA = (aX1 + b)A,where a, b ∈ C.We claim that b ∈ N (and later we prove also that a ∈ Z). To prove thiswe consider one more weight φ2 : R → Z (in fact we shall use this weightonly for monomials from R). For any F ∈ R, we denoteφ2(F ) := degt F (z, tX1, t2X2, t3X3, t−1Y0,1, . . . , t−mY0,m,t−m+1Y1,m, . . . , t−1Ym−1,m)(i.e. we assign to the variable Yu,v the weight φ2(Yu,v) := u− v).Let M ∈ R be a monomial. We readily verify that all the monomials ofAv+1(X2, X3)− 12v + 2ddYv−1,vMhave the weight φ2 strictly bigger than φ2(M). The same is true for124 (1−X1)ddY0,1M,as well as forYk+1,vddYk,vM (0 ≤ k < v − 1)and( 112(X21 −X2) ddX1+ 13 (X1X2 −X3)ddX2+ 12(X1X3 −X22) ddX3)M.The monomial z ddzM either is 0 or satisfy φ2(zddzM) = φ2(M).Multiplicity estimate 779Thus remembering the definition of D, see (2.3), we readily verify thatevery non-zero monomial appearing in DM has the weight φ2 not less thanφ2(M) and the only monomial in DM that eventually has the weight φ2equal to φ2(M) is z ddzM .Let C be the sum of monomials of A with minimal weight φ2. Then inthe r.h.s. of (2.12) the sum of monomials of minimal weight is equal to bC(all the other monomials of (aX1 + b)A obviously have a bigger value ofweight φ2).Applying to the l.h.s. of (2.12) (that is DA) the above considerations weconclude(2.13) z ddzC = bC.Comparing the coefficients on the both sides of (2.13) we find b = degz C,in particular b ∈ N.We proceed to the proof a ∈ Z. Substituting X1 = E2(z), X2 = E4(z),X3 = E6(z), Yu,v = gu,v(z), 0 ≤ u < v ≤ m in (2.12) we obtain(2.14) (aE2(z) + b)A (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))= DA (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) .We consider the Taylor expansion (at the point z = 0) of(2.15) A (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) .It is a function analytic at z = 0 because all the functions E2, E4, E6, g0,1,. . . , gm−1,m are. Let M denote the order at z = 0 of the function (2.15),that isA (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))= czM + (terms of order > M),where c 6= 0. Using (2.4) we obtainDA (z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z))= cMzM + (terms of order > M).In view of (1.1) we have E2(z) = 1+terms of order > 1. We readily deducefrom (2.14)(a+ b)czM + (terms of order > M) = cMzM + (terms of order > M).Comparing coefficients with zM on both sides in this expression (and sim-plifying out c) we readily deduce a + b = M . We have already establishedb ∈ N. Obviously, M ∈ N (as it is a degree in a Taylor series). We concludea ∈ Z.So we have established that coefficients a, b involved in (2.12) are in factintegers.780 Evgeniy ZorinNote that(2.16) D(∆−az−b) = (−aX1 − b) ∆−az−b.We denote(2.17) S (z, E2, E4, E6, g0,1, . . . , gm−1,m) :=A (z, E2, E4, E6, g0,1, . . . , gm−1,m)×∆−az−b.Applying the differential operator D to the r.h.s. of (2.17) and using (2.12),(2.16) we obtainDS = 0.Using (2.4) on the latter equality we concludeddzS(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) = 0,henceS(z, E2(z), E4(z), E6(z), g0,1(z), . . . , gm−1,m(z)) ∈ C.Recall that functions z, E2, E4, E6, g0,1, . . . , gm−1,m are all algebraicallyindependent over C, see [3] page 2. For this reason we deduceS[z,X1, X2, X3, Y ] ∈ C and therebyA = λ∆azbwith λ ∈ C. If we suppose that A is irreducible, we obtain immediately thateither (a, b) = (1, 0) or (a, b) = (0, 1). Proof of Proposition 2.1. We consider the following nested sequence of ringsC[z,X] ⊂ C[z,X, Y0,1](2.18)⊂ C[z,X, Y0,1, Y2,3]⊂ C[z,X, Y0,1, Y1,3, Y2,3]⊂ C[z,X, Y0,1, Y0,3, Y1,3, Y2,3]...⊂ C[z,X, Y0,1, . . . , Ym−3,m−2]⊂ C[z,X, Y0,1, . . . , Ym−3,m−2, Ym−1,m]⊂ C[z,X, Y0,1, . . . , Ym−3,m−2, Ym−2,m, Ym−1,m]...⊂ C[z,X, Y0,m, . . . , Ym−1,m] = R.We readily verify with the definition of D that every term Ri appearing inthe chain (2.18) satisfies DRi ⊂ Ri.Let P ⊂ R be a prime ideal of R satisfying DP ⊂ P. If P∩C[z,X] 6= {0},it contains a polynomial z∆ as shown in [5][Theorem 1.4]. So the claim ofMultiplicity estimate 781Proposition 2.1 is proved in this case. We suppose henceforth P∩C[z,X] ={0}.As we suppose P 6= {0} and P∩C[z,X] = {0}, we find in the chain (2.18)at some step an extension of rings Ri ⊂ Ri+1 satisfying P ∩ Ri = {0} andP ∩ Ri+1 6= {0}. In this case the ideal (of the ring Ri+1) P ∩ Ri+1 isa principal one, because we add exactly one variable at each step in thechain (2.18), i.e. tr.deg.RiRi+1 = 1. Hence P ∩Ri+1 is a D-stable principalideal (of the ring Ri+1, and also this ideal generates a principal D-stableideal of the ringR, becauseD-stability of a principal ideal means exactly thecondition Q|DQ on a generator of the ideal). We deduce with Lemma 2.1that z∆ ∈ P ∩Ri+1 ⊂ P, Q.E.D. References[1] P. Kozlov, On algebraic independence of functions from a certain class. Izvestiya: Mathe-matics (Izvestiya of the Russian Academy of Sciences, in russian), in press.[2] Yu. Nesterenko,Modular functions and transcendence questions. Math. Sb. 187/9 (1996),65–96 (Russian); English translation in Sb. Math. 187/9, 1319–1348.[3] Yu. Nesterenko, Some identities of Ramanujan type. Moscow Journal of Combinatoricsand Number Theory, Vol. 1 (2011), issue 2.[4] Yu. Nesterenko, Patrice Philippon (eds.), Introduction to Algebraic Independence The-ory. Lecture Notes in Mathematics, Vol. 1752, Springer, 2001.[5] F. Pellarin, La structure différentielle de l’anneau des formes quasi-modulaires pourSL2(Z). Journal de Théorie des Nombres de Bordeaux 18, no.1 (2006), 241–264.[6] P. Philippon, Indépendance algébrique et K-fonctions. J. reine angew. Math. 497 (1998),1–15.[7] P. Philippon, Une approche méthodique pour la transcendance et l’indépendance algébriquede valeurs de fonctions analytiques. J. Number Theory 64 (1997), 291–338.[8] J.-P. Serre, A Course in Arithmetic. Springer-Verlag, New York, 1973.[9] E. Zorin, Lemmes de zéros et relations fonctionnelles. Thèse de doctorat de l’UniversitéParis 6, 2010. Accessible at http://tel.archives-ouvertes.fr/tel-00558073/fr/Evgeniy ZorinDepartment of MathematicsUniversity of YorkYork, YO10 5DD, UKE-mail: evgeniyzorin@yandex.ru

Multiplicity estimate for solutions of extended Ramanujan's system

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