Selberg s integral and linear forms in zeta values

Transcription

1 Selberg s integrl nd liner forms in zet vlues Tnguy Rivol Abstrct Using Selberg s integrl, we present some new Euler-type integrl representtions of certin nerly-poised hypergeometric series. These integrls re lso shown to produce liner forms in odd nd/or even zet vlues tht generlize previous work of the uthor. 1 Selberg s integrl nd nerly-poised series Much work hs been devoted to evluting multiple hypergeometric integrls fter Beukers proof of Apéry s theorem ζ(3 is irrtionl, in which he used the following integrls equtions [Be]: nd x n (1 x n y n (1 y n (1 (1 xy n+1 dxdy = n ζ(2 + b n x n (1 x n y n (1 y n z n (1 z n (1 (1 (1 xyz n+1 dxdydz = A n ζ(3 + B n for some (explicitly computble rtionl numbers n b n, A n nd B n. See in prticulr the work of Ht [H], Rhin nd Viol [RV1, RV2], Vsilyev [V], Sorokin [So], Zudilin [Zu]. Similrly, in order to prove tht infinitely mny odd zet vlues re irrtionl, the following multiple integrl ws used in [Ri] nd [BR] : J r,n := xrn j (1 x j n [,1] (1 x +1 x dx (2r+1n+2 dx, (1 where n,, r 1 re integers such tht ( + 1n > (2r + 1n + 2. This is interesting principlly becuse there exist explicitly computble rtionl numbers (for j =,..., such tht p r,n j, J r,n = p r,n, + 1 j=2 p r,n j, ζ(j

2 nd for j 2, p r,n j, = if (n j is odd (the prity phenomenon. In [Zu], the following integrl, which generlizes Beukers integrls bove, j=1 xn j (1 x j n Q (x 1, x 2,..., x dx n+1 1 dx, [,1] with Q (x 1, x 2,..., x = 1 (1 (1 x 3 x 2 x 1, is proved to produce liner forms in odd or even zet vlues. Zudilin gives n identity between such integrls nd slight modifiction of (1, which is not t ll obvious. In this note, we construct nother kind of hypergeometric integrl, bsed on Selberg s integrl, tht lso generlizes (1 nd produces liner forms in odd or even zet vlues (Theorem 1 in 2. We will lso show how it is relted to more usul Euler-type integrls (Propositions 1 nd 2 below. We first remind the reder of the definition of hypergeometric series p F q (z : [ ] α1, α 2,..., α p pf q ; z := β 1, β 2,..., β q k= (α 1 k (α 2 k (α p k (1 k (β 1 k (β q k z k. Here, p nd q re positive integers, α j nd β j re complex numbers such tht β j N, nd (x n := x(x + 1 (x + n 1 is the Pochhmmer symbol. We lso recll well-known result of Selberg, whose proof cn be found in [Se]: if the complex numbers α, β nd γ stisfy ( 1 Re(α > 1, Re(β > 1, Re(γ > min + 1, Re(α + 1, Re(β + 1, we hve tht Sel α,β,γ := [,1] +1 x α j (1 x j β = j<l x j x l 2γ dx dx Γ(γ + jγ + 1Γ(α + jγ + 1Γ(β + jγ + 1 Γ(γ + 1Γ(α + β + ( + jγ + 2. (2 Let us now define the integrl I α,β,γ,δ (z := j (1 x j β j<l x j x l 2γ [,1] (1 zx +1 x δ+1 dx dx, (3 where z is complex number, z 1 nd α, β, γ, δ, 1 re integers ( 1 such tht ( + 1β > δ (which ensures convergence on the circle z = 1. Then the following holds. 1 More generlly, α, β nd γ could be tken to be suitble complex numbers. 2

3 Proposition 1. Under the bove conditions, I α,β,γ,δ (z = Sel α,β,γ +2F +1 [ δ + 1, α + 1, α + γ + 1, α + 2γ + 1,..., α + γ + 1 α + β + 2γ + 2, α + β + (2 1γ + 2,..., α + β + γ + 2 ; z ]. (4 The hypergeometric function +2 F +1 on the RHS of (4 is sid to be nerlypoised becuse the sum of n upper prmeter (other thn δ + 1 nd suitble lower prmeter is invrint : for ll j =,...,, (α + jγ (α + β + (2 jγ + 2 = 2α + β + 2γ + 3. Proof. This cn be cn thought of s generting function rewriting of Selberg s identity. To simplify, we write c α,β,γ,δ := Then, using the expnsion Γ(γ + jγ + 1Γ(β + jγ + 1 Γ(δ + 1Γ(γ (1 t = δ+1 k= ( δ + k nd interverting the signs in (3, we get I α,β,γ,δ = (z ( δ + k z k k= = c α,β,γ,δ k k= [,1] +1 Γ(k + δ + 1 Γ(k + 1 k t k x α+k j (1 x j β ( j<l Γ(k + α + jγ + 1 Γ(k + α + β + ( + jγ + 2 x j x l 2γ dx dx z k, (5 where we hve used Selberg s identity (2 with α + k replcing α in the lst step. We now note tht, thnks to the identity (x n = Γ(x + n/γ(x, the eqution (5 is simply the RHS of (4. An interesting consequence of Proposition 1 is the construction of new integrl representtions for I α,β,γ,δ (z in which the discriminnt j<l x j x l does not pper. Proposition 2. Let σ be ny permuttion of the set {,..., }. Then I α,β,γ,δ (z = ( Γ(γ + jγ + 1Γ(β + jγ + 1 Γ(γ + 1Γ(β + ( j + σ(jγ + 1 [,1] +1 xα+jγ j (1 x j β+( j+σ(jγ (1 zx x δ+1 dx dx. (6 3

4 Proof. We first note tht in (4 n upper prmeter (other thn δ + 1 is lwys less thn lower prmeter, tht is to sy, for ny j, k, α + jγ + 1 < α + β + ( + kγ + 2. This inequlity cn be reformulted s follows : for ny permuttion σ of the set {,..., } nd for ny j, the inequlity α + jγ + 1 < α + β + ( + σ(jγ + 2 holds. Hence, we cn pply clssicl identity of Euler which expresses hypergeometric series s multiple integrl (cf. [Sl], p. 18 I α,β,γ,δ (z = ( Γ(γ + jγ + 1Γ(β + jγ + 1Γ(α + β + ( + σ(jγ + 2 Γ(γ + 1Γ(β + ( j + σ(jγ + 1Γ(α + β + ( + jγ + 2 j (1 x j β+( j+σ(jγ [,1] +1 (1 zx x δ+1 dx dx. (7 To conclude, we note tht, for ny permuttion σ, Γ(α + β + ( + σ(jγ + 2 = Γ(α + β + ( + jγ + 2, which simplifies the Gmm quotient in (7 nd proves the identity (6. 2 The well-poised cse Specil ttention should be pid to prticulr cse of Proposition 1 : when δ = 2α + β + 2γ + 1, then the hypergeometric function on the RHS of (4 is well-poised. In this cse, the corresponding integrl cn be evluted t z = 1 in term of odd (resp. even zet vlues, ccording to the vlues of the prmeters. (In the nerly-poised cse, such decomposition involves both the even nd odd zet vlues. Theorem 1. For integers α, β, γ nd 1 such tht β > 2α + 2γ + 2, the integrl J α,β,γ := xα j (1 x j β j<l x j x l 2γ dx [,1] (1 x +1 x 2α+β+2γ+2 dx (8 is convergent nd there exist explicitly computble rtionl numbers P α,β,γ j, j =,..., such tht (for J α,β,γ = P α,β,γ, + j=2 P α,β,γ j, ζ(j. (9 Furthermore, for j 2, P α,β,γ j,n = if (β j is odd. 4

5 Clerly, when γ =, the integrl J α,β,γ in the introduction s prticulr cse. includes the integrl J r,n mentioned Proof. It is now well-estblished tht the prity phenomenon is consequence of the well-poisedness of the underlying hypergeometric function (see [RZ] for more detils of this ide. We will therefore simply sketch the resoning nd give references t ech step. With the nottion used in the proof of Proposition 1 nd with δ = 2α + β + 2γ + 1, trivil trnsformtion of the Pochhmmer symbols of the hypergeometric series (4 gives the following identity : J α,β,γ = c α,β,γ,δ k= (k + 1 δ (k + α + jγ + 1. (1 β+γ+1 The expnsion in prtil frctions of the rtionl function R(X := (X + 1 δ (X + α + jγ + 1 β+γ+1 immeditely implies (9 (see [Ri], [BR]. Furthermore, pplying the trivil identity (x n = ( 1 n ( x n+1 n to R(X, we obtin the crucil symmetry reltion R( X δ 1 = ( 1 (β+1 R(X. (11 From (11 (essentilly by the uniqueness of the decomposition in prtil frctions, we then deduce tht P α,β,γ j, = if j 2 nd j + (β + 1 is odd, which completes the proof (see [Fi], [Co], or [Ri], [BR] for the slightly different originl rgument. An interesting problem would be to prove the identity (9 nd the prity phenomenom without expnding the integrl (8 s the series (1 (the method used in [RV1, RV2] could be relevnt here. Furthermore, hopefully we will find new diophntine pplictions of Theorem 1, other thn those lredy known for γ =. Acknowledgment. I thnk F. Amoroso for suggesting the ide of using Selberg s integrl to produce new rtionl liner forms in zet vlues. References [Be] F. Beukers, A note on the irrtionlity of ζ(2 nd ζ(3, Bull. London Mth. Soc. 11 (1979, no. 3, [BR] K. Bll, T. Rivol, Irrtionlité d une infinité de vleurs de l fonction zêt ux entiers impirs, Invent. Mth. 146 (21, no. 1,

6 [Co] P. Colmez, Arithmétique de l fonction zêt, Actes des journées X-UPS 22 L fonction zêt. [Fi] S. Fischler, Irrtionlité de vleurs de zêt (d près Apéry, Rivol,..., Séminire Bourbki 22-23, exposé numéro 91, novembre 22. [H] M. Ht, A new irrtionlity mesure for ζ(3, Act Arith. 92 (2, no. 1, [RV1] G. Rhin, C. Viol, On permuttion group relted to ζ(2, Act Arith. 77 (1996, no. 1, [RV2] G. Rhin, C. Viol, The group structure for ζ(3, Act Arith. 97 (21, no. 3, [Ri] T. Rivol, L fonction zêt de Riemnn prend une infinité de vleurs irrtionnelles ux entiers impirs, C. R. Acd. Sci. Pris Sér. I Mth. 331 (2, no. 4, [RZ] T. Rivol, W. Zudilin, Diophntine properties of numbers relted to Ctln s constnt, to pper in Mth. Annlen (23. [Se] A. Selberg, Remrks on multiple integrl (Norwegin, Norsk Mt. Tidsskr. 26 (1944, [Sl] L.J. Slter, Generlized Hypergeometric Functions, Cmbridge University Press, [So] V.N. Sorokin, On Apéry theorem (Russin. Russin summry, Vestnik Moskov. Univ. Ser. I Mt. Mekh. 1998, no. 3, 48 53, 74; trnsltion in Moscow Univ. Mth. Bull. 53 (1998, no. 3, [V] D.V. Vsilyev, On smll liner forms for the vlues of the Riemnn zetfunction t odd points, Preprint no. 1 (558, Nt. Acd. Sci. Belrus, Institute Mth., Minsk (21. [Zu] W. Zudilin, Well-poised hypergeometric service for diophntine problems of zet vlues, Actes des 12èmes rencontres rithmétiques de Cen (June 29 3, 21, to pper in J. Théorie Nombres Bordeux (23. Tnguy Rivol Lbortoire de Mthémtiques Nicols Oresme CNRS UMR 6139 Université de Cen BP Cen cedex, Frnce rivol@mth.unicen.fr 6