On a permutation group related to ζ(2)

Transcription

1 ACTA ARITHMETICA LXXVII. (996) On a permutation group related to ζ(2) by Georges Rhin (Metz) Carlo Viola (Pisa). Introduction. Ever since the appearance of Beukers paper [], it was clear that double integrals of the type \ \ dx dy (.) P (x, y) xy, where P (x, y) Z[x, y], are relevant to the study of Diophantine properties of ζ(2) n n 2 π 2 /6, although the problem of finding polynomials P (x, y) yielding good irrationality measures of ζ(2) proved to be a difficult one. With the choice P (x, y) L n (x)( y) n, where L n (x) d n n! dx n (xn ( x) n ) is the nth Legendre polynomial, Beukers obtained the sequence of rational approximations to ζ(2) previously found by Apéry through an entirely different method, hence gave a new proof of Apéry s result that log log 2 2 is an irrationality measure of ζ(2). We recall that λ is an irrationality measure of the irrational number α if for any ε > 0 there exists a constant q 0 q 0 (ε) > 0 such that α p q > q λ ε for all integers p q with q > q 0. We denote by µ(α) the minimum of such exponents λ. [23]

2 24 G. Rhin C. Viola By repeated partial integration, one sees that \ \ n dx dy (.2) L n (x)( y) xy \ \ ( x( x)y( y) ( ) n xy ) n dx dy xy, the left side of (.2) is easily seen to be a n b n ζ(2), with b n integer a n a rational number having a controlled denominator, while the right side of (.2) is suitable to get asymptotic estimates of a n b n ζ(2) b n as n. The search for better polynomials P (x, y) in (.) was not pursued until the appearance of the paper [3] by Dvornicich Viola, where the irrationality measure µ(ζ(2)) < was obtained. This was later improved to µ(ζ(2)) < by Hata [5] to µ(ζ(2)) < by Rhin Viola [8]. In a recent paper, Hata [7] considered the integral \ \ ( x 5 ( x) 5 y 4 ( y) 4 ) n dx dy (.3) ( xy) 2 xy, transformed it into an integral of the type (.) by 2n-fold partial integration, thus proving that (.3) has indeed the form a n b n ζ(2) with a n Q b n Z. By making use of the p-adic valuation of suitable binomial coefficients, Hata showed that the denominator of a n is relatively small, proved the remarkable result µ(ζ(2)) < He subsequently improved this to µ(ζ(2)) < ([7], Addendum) by combining his method with the properties of a birational transformation of the plane introduced in [8]. The purpose of the present paper is the arithmetical study of a family of integrals generalizing (.3). We define \ \ x h ( x) i y k ( y) j dx dy (.4) I(h, i, j, k, l) ( xy) i+j l xy, where h, i, j, k, l are any non-negative integers, we show that the Diophantine properties of the integrals (.4) rest upon an underlying algebraic structure. Specifically, we consider the action on the integrals (.4) of a naturally arising permutation group. Some algebraic properties of the integrals (.4) (in the more general case where h, i, j, k, l are any complex parameters with real parts > ) were studied in 905 by A. C. Dixon [2] who, however, gave no arithmetical applications of them. Dixon found that the value of (.4) is unchanged under a cyclic permutation of h, i, j, k, l. This can be shown e.g. by applying to the

3 Permutation group related to ζ(2) 25 integral (.4) the birational transformation ξ x (.5) τ : xy, η xy, defined in [8]. Moreover, Dixon remarked that the Euler integral representation of Gauss s hypergeometric function shows that the quantity (.6) I(h, i, j, k, l) Γ (h + )Γ (i + )Γ (j + )Γ (k + )Γ (l + ) is a symmetric function of the sums h + i, i + j, j + k, k + l, l + h. In our context the gamma-factors are factorials, they provide information on the p-adic valuation of the rational part a n of \ \ ( x h ( x) i y k ( y) j ) n dx dy (.7) I(hn, in, jn, kn, ln) ( xy) i+j l xy a n b n ζ(2). A crucial step to obtain this is to characterize in terms of quotients of factorials the (left) cosets of a subgroup T, related to the transformation (.5), in a permutation group Φ related to the hypergeometric integral transformation ϕ which leaves the value of (.6) unchanged, we do this in Section 3. We remark that if i + j l > min{h, i, j, k}, (.4) cannot be transformed by partial integration into an integral of the type (.) to which Hata s arithmetical method [7] can be successfully applied, in fact if i + j l > min{h, k}, by the partial integration method it is not even clear that (.4) equals a bζ(2) for some a Q b Z. Therefore, in order to apply for any admissible choice of h, i, j, k, l the whole group Φ to the study of the p-adic valuation of the rational part a n of (.7), we need first to get rid of the partial integration method, to show that (.4) has the form a bζ(2) without using a representation of the type (.). In Section 2 we achieve this, we find an (optimal) pair of non-negative integers M, N such that d M d N a Z (we denote d m l.c.m.{, 2,..., m}). The choice h i 2, j k 4, l 3 in (.7) allows us to prove the irrationality measure µ(ζ(2)) < We incidentally remark that the one-dimensional analogue of our method applies to the integrals \ ( x h ( x) i ) n dx ( + x) i l + x, 0 where h, i, l are any integers satisfying h > max{0, l}, i > max{0, l}, allows one to obtain irrationality measures of log 2. Here the choice

4 26 G. Rhin C. Viola h i 7, l yields the same sequence of rational approximations to log 2 found by Rukhadze [9] (see also Hata [5]), hence the bound µ(log 2) < This paper is organized as follows. In Section 2 we simplify generalize a method introduced in our previous paper ([8], Section 3). Thus we prove some arithmetical lemmas showing that for any non-negative integers h, i, j, k, l we have I(h, i, j, k, l) a bζ(2) with b Z d M d N a Z, where M N are invariant under the action of the transformation (.5) on I(h, i, j, k, l), allowing one to express the integer b as a double contour integral. In Section 3 we apply to I(h, i, j, k, l) the hypergeometric transformation ϕ, we can do this by imposing the restriction that not only (.8) h, i, j, k, l but also (.9) j + k h, k + l i, l + h j, h + i k, i + j l are non-negative. Denoting by σ the transformation that interchanges the variables x, y in the integral I(h, i, j, k, l), we consider the permutations ϕ, τ σ of the ten integers (.8) (.9) induced by the integral transformations ϕ, τ σ respectively, we characterize the left cosets of the subgroup T τ, σ in the group Φ ϕ, τ, σ in terms of quotients of factorials. In Section 4 we analyse the p-adic valuation hence the denominator of a n by a discussion close to Hata s method [7]. Our approach differs from Hata s in the use of the p-adic valuation of the factorials occurring in the transformation formulae corresponding to the left cosets of T in Φ, in place of Hata s analysis of the p-adic valuation of binomial coefficients. In Section 5 we give the asymptotic estimates of I(hn, in, jn, kn, ln) a n b n ζ(2) of b n as n, under the further restriction that the integers (.8) (.9) are strictly positive, we prove the irrationality measure µ(ζ(2)) < mentioned above. We point out here that, even though the values of the integers (.8) in Section 5 must be such that (.8) (.9) are strictly positive, we cannot impose this restriction from the very beginning, since our arithmetical lemmas in Section 2 consist in reducing I(h, i, j, k, l) to simpler integrals for which some of the integers (.8) vanish, or some of (.9) may be negative. Hence it is essential to restrict successively the signs of (.8) (.9) as indicated above.

5 Permutation group related to ζ(2) 27 We wish to make a concluding remark on the motivation of this paper. Unlike Diophantine approximation to algebraic numbers, where fairly general methods are available even in the search for effective irrationality measures, the Diophantine theory for transcendental constants is characterized by lack of generality in the tools employed. However, the algebraic approach we work out in this paper also applies to the Diophantine study of a class of constants other than ζ(2) or log 2, which can be related to suitable integrals. We hope that our paper will be primarily viewed as a contribution to the quest for general methods in the theory of Diophantine approximation to transcendental (or conjecturally transcendental) constants. We are indebted to F. Amoroso, R. Dvornicich D. Masser for interesting discussions on some aspects of our results. 2. Arithmetical lemmas. Let h, i, j, k, l be integers. We consider the integral \ \ x h ( x) i y k ( y) j dx dy (2.) I(h, i, j, k, l) ( xy) i+j l xy, note that the condition for I(h, i, j, k, l) to be finite is that h, i, j, k, l are all non-negative, which we shall henceforth assume. We recall from [8] that the transformation τ defined by ξ x τ : xy, η xy, has period 5 maps the unit square (0, ) (0, ) onto itself. Moreover, the function x( x)y( y) xy the measure dx dy xy are invariant under the action of τ. Let J 0 I(h, i, j, k, l), let J be the integral obtained by applying to J 0 the transformation τ, i.e. by making in (2.) the change of variables x ξη, τ : y η ξη, then replacing ξ, η with x, y respectively. We find J I(i, j, k, l, h), so that with the action of τ on J 0 we associate the cyclic permutation τ of h, i, j, k, l represented by the diagram

6 28 G. Rhin C. Viola (2.2) As usual, we denote this permutation by τ (h i j k l). Thus, if we call J m the integral obtained by applying τ m to J 0, we have (2.3) J 0 I(h, i, j, k, l), J I(i, j, k, l, h), J 2 I(j, k, l, h, i), J 3 I(k, l, h, i, j), J 4 I(l, h, i, j, k). Similarly, by applying to I(h, i, j, k, l) the transformation { ξ y, σ : η x, i.e. by interchanging the variables x, y in the integral, we get I(k, j, i, h, l). Hence with the action of σ on I(h, i, j, k, l) we associate the permutation σ (h k)(i j) that interchanges h with k i with j. Thus σ changes the diagram (2.2) by the symmetry about its vertical midline. Therefore the permutation group τ, σ generated by τ σ is isomorphic to the dihedral group D 5 of order 0, the value of I(h, i, j, k, l) is invariant under the action of τ, σ. Let now d n l.c.m.{, 2,..., n}, d 0 for completeness. Our aim is to show that for any non-negative integers h, i, j, k, l we have I(h, i, j, k, l) a bζ(2) with a Q, b Z, to find two non-negative integers M, N, as small as possible, such that d M d N a Z, thus generalizing Hata s method [7]. The above a b are the same for the ten integrals obtained by the action of τ, σ on I(h, i, j, k, l). Accordingly, M N will be defined to be invariant under the actions of τ σ (see Theorem 2.2 below).

7 (2.4) Permutation group related to ζ(2) 29 Theorem 2.. Let h, i, j, k, l be non-negative integers, let Then the integral satisfies M 0 max{k + l i, l + h j, i + j l}, N 0 max{j + k h, min{k + l i, l + h j}, h + i k}. \ \ J 0 with b Z d M0 d N0 a Z. x h ( x) i y k ( y) j ( xy) i+j l dx dy xy J 0 a bζ(2) R e m a r k 2.. Our assumption h, i, j, k, l 0 obviously implies M 0, N 0 0, even if some of the integers are negative. j + k h, k + l i, l + h j, h + i k, i + j l The proof of Theorem 2. is based on some arithmetical lemmas. Lemma 2. (Beukers []). If i + j l 0, then Theorem 2. holds. P r o o f. If i + j l < 0, since \ \ x r y s dx dy (r + )(s + ) for r, s 0, we have J 0 Q, d k+l i d l+h j J 0 Z, M 0 max{k + l i, l + h j}, N 0 min{k + l i, l + h j}. If i + j l 0, by Lemma. in [7] we have J 0 a bζ(2) with b Z d M d N a Z, where M max{h + i, j + k} M 0 N min{max{h + i, j + k h}, max{j + k, h + i k}} max{j + k h, min{j + k, h + i}, h + i k} N 0. Lemma 2.2. If j k 0, then Theorem 2. holds. P r o o f. By (2.3) we have J 0 I(h, i, 0, 0, l) J 2 I(0, 0, l, h, i), M 0 max{l + h, i l}, N 0 max{l i, h + i}.

8 30 G. Rhin C. Viola Denoting by M 2 N 2 the corresponding integers for the integral J 2, i.e. the integers defined by (2.4) with h, i, j, k, l replaced by j, k, l, h, i respectively, we have in the present case M 2 max{h + i, l i} N 0, N 2 max{l + h, i l} M 0. Hence the conclusion follows by applying Lemma 2. to J 0 if i l 0, or to J 2 if i l 0. Lemma 2.3. If k l 0, then Theorem 2. holds. P r o o f. Again by (2.3) we have J 0 I(h, i, j, 0, 0) J I(i, j, 0, 0, h), M 0 max{h j, i + j}, N 0 max{j h, h + i}. Denoting by M, N the integers defined by (2.4) with h, i, j, k, l replaced by i, j, k, l, h respectively, we now find M max{h + i, j h} N 0, N max{h j, i + j} M 0. Hence the conclusion follows by applying Lemma 2.2 to J. Lemma 2.4. If i k 0, then Theorem 2. holds. P r o o f. If h 0, we apply Lemma 2.2 after interchanging x y in J 0. If l 0, we apply Lemma 2.3. If j l 0, we apply Lemma 2.. Hence we may assume (2.5) min{h, l, j l} > 0. Then we have the decomposition \ \ x h ( y) j dx dy J 0 ( xy) j l xy \ \ ( x h ( y) j ( xy) j l xh ( y) j ( xy) j l + xh ( y) j ( xy) j l ) dx dy xy I(h, 0, j, 0, l ) I(h, 0, j, 0, l ) + I(h, 0, j, 0, l) J () 0 J (2) 0 + J (3) 0, say, with J (r) 0 I(h r, 0, j r, 0, l r ). For each r, r 3, at least one of the integers h r, l r, j r l r is smaller than the corresponding integer associated with J 0, while none is larger. Moreover, if we denote by M (r) 0, N (r) 0 the integers defined by (2.4) with h, i, j, k, l replaced by h r, 0, j r, 0, l r respectively, we see that M (r) 0, N (r) 0 do not exceed the corresponding integers M 0, N 0 for J 0. By iterating the above decomposition for each integral J (r) 0 satisfying

9 Permutation group related to ζ(2) 3 (2.5), i.e. such that min{h r, l r, j r l r } > 0, we express J 0 as a linear combination with integer coefficients of finitely many integrals J (s) 0 a s b s ζ(2) satisfying b s Z d M0 d N0 a s Z. Lemma 2.5. If k 0, then Theorem 2. holds. P r o o f. If i 0 or j 0 or l 0, we apply Lemmas 2.4, 2.2 or 2.3 respectively. If i + j l 0, we apply Lemma 2.. Hence we may assume (2.6) min{i, j, l, i + j l} > 0. Then we iterate the decomposition \ \ x h ( x) i ( y) j dx dy J 0 ( xy) i+j l xy \ \ ( x h ( x) i ( y) j ( xy) i+j l + xh ( x) i ( y) j ( xy) i+j l xh ( x) i ( y) j ) dx dy ( xy) i+j l xy I(h, i, j, 0, l ) + I(h, i, j, 0, l ) I(h, i, j, 0, l ) for each integral satisfying (2.6), we conclude as in the proof of Lemma 2.4. P r o o f o f T h e o r e m 2.. If h 0 or k 0 we apply Lemma 2.5, possibly after interchanging x y. If i + j l 0 we apply Lemma 2.. Hence we may assume (2.7) min{h, k, i + j l} > 0. Then we iterate the decomposition \ \ x h ( x) i y k ( y) j dx dy J 0 ( xy) i+j l xy \ \ ( x h ( x) i y k ( y) j ( xy) i+j l xh ( x) i y k ( y) j ( xy) i+j l I(h, i, j, k, l) I(h, i, j, k, l + ) ) dx dy xy for each integral satisfying (2.7), we conclude as in the proof of Lemma 2.4. The integers M 0, N 0 defined by (2.4) are invariant under the action of σ, but not of τ. For each m, 0 m 4, let M m, N m be the corresponding integers for the integral J m in (2.3), i.e. those defined by (2.4) with h, i, j,

10 32 G. Rhin C. Viola k, l replaced by τ m (h), τ m (i), τ m (j), τ m (k), τ m (l) respectively, where τ is the permutation (2.2). Since J 0... J 4, for any m q with 0 m 4, 0 q 4, the pair M m, N m is admissible for J q. In other words, we have J q a bζ(2) with b Z d Mm d Nm a Z. Thus we seek the best pairs among M m, N m (0 m 4), i.e. those for which M m + N m is minimal. We let τ σ act on the five integers (2.8) j + k h, k + l i, l + h j, h + i k, i + j l by defining τ (j + k h) τ (j) + τ (k) τ (h) k + l i, σ(j + k h) σ(j) + σ(k) σ(h) h + i k, so on. Hence the cyclic permutation (j + k h k + l i l + h j h + i k i + j l) is also called τ, similarly (j + k h h + i k)(k + l i l + h j) is called σ. We define M max{j + k h, k + l i, l + h j, h + i k, i + j l}, (2.9) N max{τ (M), min{τ 2 (M), τ 3 (M)}, τ 4 (M)}. If two (or more) of the five integers (2.8) are maximal, the definition of N, a priori, may depend on the choice of the maximal integer M among (2.8). However, the proof of Theorem 2.2 will show that N is well defined, for if M M are both maximal, then the N above N max{τ (M ), min{τ 2 (M ), τ 3 (M )}, τ 4 (M )} are equal. Moreover, the M, N defined by (2.9) are clearly invariant under the actions of τ σ. Theorem 2.2. Let h, i, j, k, l be non-negative integers, let the integrals J q (0 q 4) be as in (2.3), let M, N be defined by (2.9). For any q we have J q a bζ(2) with b Z, d M d N a Z, M, N is the best among the (unordered) pairs M m, N m (0 m 4). P r o o f. We shall show that two cases may occur: either the five pairs M m, N m are all equal (up to the interchange of M m N m ) to the pair M, N, or three of them are equal to M, N the remaining two are equal

11 Permutation group related to ζ(2) 33 to a pair worse than M, N. Thus in either case M, N is the best pair for each J q. We may assume with no loss of generality that M i+j l. For otherwise we should replace J 0 with the integral J r such that M τ r (i + j l), accordingly each J q with the integral J s such that s r + q (mod 5). In each of the five pairs M m, N m the maximal integer i + j l occurs either in the expression for M m or for N m, so that one element of the best pair must be M i + j l. To find the other, we consider the different possibilities for the maximum of the remaining four integers. Note that the cases max{j + k h, k + l i, l + h j, h + i k} k + l i max{j + k h, k + l i, l + h j, h + i k} l + h j are changed into each other by σ, so that the same discussion applies to both, similarly for the cases max{j + k h, k + l i, l + h j, h + i k} j + k h max{j + k h, k + l i, l + h j, h + i k} h + i k. Therefore, up to the action of σ, we have only twelve distinct possibilities for the ordering of the integers (2.8): (2.0) (2.) (2.2) (2.3) j + k h l + h j h + i k k + l i i + j l, l + h j j + k h h + i k k + l i i + j l, j + k h h + i k l + h j k + l i i + j l, h + i k j + k h l + h j k + l i i + j l, l + h j h + i k j + k h k + l i i + j l, h + i k l + h j j + k h k + l i i + j l, k + l i l + h j h + i k j + k h i + j l, l + h j k + l i h + i k j + k h i + j l, k + l i h + i k l + h j j + k h i + j l, h + i k k + l i l + h j j + k h i + j l, l + h j h + i k k + l i j + k h i + j l, h + i k l + h j k + l i j + k h i + j l. In the five pairs M m, N m the smaller of M m N m is h + i k three times k + l i twice if we are in one of the cases (2.0); l + h j three times

12 34 G. Rhin C. Viola k + l i twice in (2.); j + k h three times k + l i twice in (2.2). In any of the cases (2.3) we find j + k h all the five times. Hence in any of the above cases the best choice for the smaller of M m N m (m 0,..., 4) is N max{j + k h, min{k + l i, l + h j}, h + i k}. This proves Theorem 2.2. Under the assumptions of Theorems 2. or 2.2, the integer b can be expressed as a double contour integral. Lemma 2.6. Let h, i, j, k, l be non-negative integers. If \ \ J 0 x h ( x) i y k ( y) j ( xy) i+j l dx dy xy a bζ(2) with a Q b Z, then, for any ϱ, ϱ 2 > 0, b \ \ xh ( x)iyk ( y)j dx dy 4π 2 ( xy) i+j l xy, C C x where C {x C : x ϱ } C x {y C : y /x ϱ 2 }. P r o o f. As in the proof of Theorem 2., we can express J 0 as a linear combination with integer coefficients of integrals having either i + j l 0, or, possibly after interchanging x y, k 0. If k 0, by the proofs of Lemmas we express the integral as a linear combination with integer coefficients of integrals either having i + j l 0, or such that at least two consecutive integers among h, i, j, k, l vanish. In the latter case, by applying a suitable power of the transformation τ we may assume j k 0, then the proof of Lemma 2.2 shows that the integral either has i+j l 0 or can be transformed into one having i+j l 0. If i+j l < 0, the integral is a rational number. Therefore where β t Z J 0 T β t J (t) + (rational number), t \ \ (2.4) J (t) x h t ( x) i t y k t ( y) j t Thus, if we put (2.5) J (t) a (t) b (t) ζ(2) dx dy xy.

13 with a (t) Q b (t) Z, we get (2.6) b Permutation group related to ζ(2) 35 T β t b (t). t We now apply to the double contour integral J 0 \ \ xh ( x)iyk ( y)j dx dy 4π 2 ( xy) i+j l xy C C x the same linear decompositions the same transformations used for J 0. Since τ changes \ \ (2.7) xh ( x)iyk ( y)j dx dy ( xy) i+j l xy C C x into \ y ϱ ϱ 2 which clearly equals \ \ xi ( x)jyl ( y)k dx dy ( xy) j+k h xy, x /y /ϱ 2 x ϱ y /x ϱ 2 \ x i ( x) j y l ( y) k ( xy) j+k h dx dy xy for any ϱ, ϱ 2, ϱ, ϱ 2 > 0, since (2.7) vanishes if i + j l < 0, we obtain T (2.8) J0 β t J (t), t with the same β t Z as above with (2.9) J (t) \ \ dx dy 4π 2 x ht ( x) it y kt ( y) jt xy. C C x Let now P (x, y) δ δ2 r0 s0 α rsx r y s be any polynomial with α rs Z. Then (see [7], Lemma.) \ \ δ δ dx dy 2 P (x, y) xy \ \ α rs x r s dx dy y xy U V ζ(2), r0 s0 with U Q (denoting here i ) V min{δ,δ 2 } 2πi r0 \ α rr δ δ 2 r0 s0 z ϱ P (z, /z) dz z. α rs 2πi \ z ϱ z r s dz

14 36 G. Rhin C. Viola For any z 0 we have \ 2πi Hence y /z ϱ 2 P (z, y) zy dy z V 2πi \ z ϱ 4π 2 \ z ϱ 2πi ( 2πi \ P (z, y) y /z dy z y /z ϱ 2 \ P (z, y) zy dy y /z ϱ 2 \ P (z, y) zy y /z ϱ 2 dy dz. ) dz P (z, /z). Taking P (x, y) x h t ( x) i t y k t ( y) j t, from (2.4), (2.5) (2.9) we get b (t) \ \ dx dy 4π 2 x ht ( x) it y kt ( y) jt xy J (t). y /x ϱ 2 x ϱ Therefore, by (2.6) (2.8), b T β t b (t) t T β t J (t) J 0. t 3. The hypergeometric permutation. Let α, β, γ be complex parameters, γ 0,, 2,..., y a complex variable satisfying y <. The Gauss hypergeometric function F (α, β; γ; y) 2 F (α, β; γ; y) is defined by (3.) F (α, β; γ; y) n0 (α) n (β) n (γ) n yn n!, where (α) 0, (α) n α(α + )... (α + n ) (n, 2,...), similarly for (β) n (γ) n. By Euler s integral representation we have, for Re γ > Re β > 0, F (α, β; γ; y) Γ (γ) \ x β ( x) γ β Γ (β)γ (γ β) ( xy) α dx 0 ([4], p. 59), this gives the analytic continuation of F (α, β; γ; y). Since, by (3.), F (α, β; γ; y) F (β, α; γ; y), if Re γ > max{re α, Re β} min{re α, Re β} > 0 we obtain

15 (3.2) Permutation group related to ζ(2) 37 \ x β ( x) γ β Γ (β)γ (γ β) ( xy) α dx 0 \ x α ( x) γ α Γ (α)γ (γ α) ( xy) β dx. 0 Here in the sequel we choose five non-negative integers h, i, j, k, l such that j +k h, k +l i, l+h j, h+i k, i+j l are also non-negative. Taking in (3.2) we get \ h!i! 0 α i + j l +, β h +, γ h + i + 2, x h ( x) i dx ( xy) i+j l+ \ x i+j l ( x) l+h j (i + j l)!(l + h j)! ( xy) h+ dx. 0 Multiplying by y k ( y) j integrating in 0 y we obtain, by (2.), (3.3) I(h, i, j, k, l) h!i! I(i + j l, l + h j, j, k, l). (i + j l)!(l + h j)! Dividing (3.3) by j!k!l! we have (3.4) I(h, i, j, k, l) h! i! j! k! l! I(i + j l, l + h j, j, k, l) (i + j l)!(l + h j)!j!k!l!. Let ϕ be the hypergeometric integral transformation acting on (3.5) I(h, i, j, k, l) h! i! j! k! l! as is described above. It is natural to associate with the action of ϕ on (3.5) a permutation ϕ of the integers h, i, j, k, l, j + k h, k + l i, l + h j, h+i k, i+j l which we define to be the following product of transpositions (i.e. 2-cycles): Note that ϕ (h i + j l)(i l + h j)(j + k h k + l i). ϕ(j + k h) k + l i j + k (i + j l) ϕ(j) + ϕ(k) ϕ(h), similarly ϕ(k + l i) ϕ(k) + ϕ(l) ϕ(i), etc. In accordance with

16 38 G. Rhin C. Viola Section 2, we call τ σ the permutations τ (h i j k l)(j + k h k + l i l + h j h + i k i + j l), σ (h k)(i j)(j + k h h + i k)(k + l i l + h j). We are interested in the structure of the permutation group Φ ϕ, τ, σ generated by ϕ, τ σ. We have already remarked that the subgroup T τ, σ is isomorphic to the dihedral group D 5 of order 0, it is easy to check that ϕ, σ is isomorphic to the dihedral group D 6 of order 2. Following a remark of Dixon [2], we note that ϕ, τ σ can be viewed as three permutations of five integers only, i.e. of the five sums h + i, i + j, j + k, k + l, l + h, by defining ϕ(h + i) ϕ(h) + ϕ(i), etc. In fact, we have the following decompositions into cycles: ϕ (i + j l + h), τ (h + i i + j j + k k + l l + h), σ (h + i j + k)(k + l l + h). Since the symmetric group S 5 of the 5! 20 permutations of five elements is generated by a cyclic permutation of the five elements a transposition, we see that Φ ϕ, τ, σ ϕ, τ is isomorphic to S 5. Moreover, the value of (3.5) is invariant under the action of Φ, whence (3.5) is a symmetric function of the sums h + i, i + j, j + k, k + l, l + h. Since Φ 20 T 0, there are 2 left cosets of T in Φ. Each left coset can be characterized in terms of the factorials occurring in the corresponding transformation formulae for I(h, i, j, k, l) such as (3.3). To see this, note that from our definitions it follows that if we apply to (3.5) any product χ of integral transformations ϕ, τ σ, we obtain I(χ(h), χ(i), χ(j), χ(k), χ(l)), χ(h)!χ(i)!χ(j)!χ(k)!χ(l)! where χ is the corresponding product of permutations ϕ, τ σ in reverse order. In other words, the above mapping χ χ between the groups Φ ϕ, τ, σ Φ ϕ, τ, σ is an anti-isomorphism. For instance, if we apply τϕ (i.e. first ϕ then τ) to (3.5) we get I(h, i, j, k, l) h!i!j!k!l! I(i + j l, l + h j, j, k, l) I(l + h j, j, k, l, i + j l) (i + j l)!(l + h j)!j!k!l! (l + h j)!j!k!l!(i + j l)!,

17 Permutation group related to ζ(2) 39 if we apply ϕτ (first τ then ϕ) to (h, i, j, k, l) we find (h, i, j, k, l) (i, j, k, l, h) (l + h j, j, k, l, i + j l). Therefore, if for any ϱ T τ, σ we apply ϕϱ to (h, i, j, k, l), we change both the numerator the denominator of the right side of (3.4) by a suitable permutation of i+j l, l +h j, j, k, l. If the numerator is changed into an integral Ĩ, we obtain h! i! I(h, i, j, k, l) I(i + j l, l + h j, j, k, l) (i + j l)!(l + h j)! h! i! (i + j l)!(l + h j)!ĩ. Thus we find the same factor h! i! (3.6) (i + j l)!(l + h j)! for each of the ten elements of the left coset ϕt. Plainly the same argument applies to every left coset of T in Φ. Since every element of Φ ϕ, τ is a suitable product of permutations each equal to ϕ or to τ, since the value of (3.5) is invariant under the action of Φ moreover h + i (i + j l) + (l + h j), we see that in the transformation formulae for I(h, i, j, k, l) each factor of the type (3.6) corresponding to a left coset of T in Φ is a quotient of factorials satisfying the following properties: (i) The numerator the denominator are products of the same number of factorials. (ii) The integers occurring in the numerator belong to the set {h, i, j, k, l} the integers in the denominator belong to {j + k h, k + l i, l + h j, h + i k, i + j l}. (iii) The sum of the integers in the numerator equals the sum of the integers in the denominator. It is easy to see that the following elements of Φ: (3.7) ι, ϕ, τ ϕ, τ 2 ϕ, τ 3 ϕ, τ 4 ϕ, ϕτ ϕ, ϕτ 2 ϕ, ϕτ 3 ϕ, ϕτ 4 ϕ, τ 2 ϕτ ϕ, ϕτ 2 ϕτ ϕ, where ι denotes the identity, yield distinct factors of the type (3.6), hence are pairwise left-inequivalent mod T, i.e. representatives of all the 2 left cosets of T in Φ. Let the permutations (3.7) be denoted by χ, χ 2,..., χ 2 respectively. The corresponding transformation formulae for I(h, i, j, k, l) are the following:

18 40 G. Rhin C. Viola I(h, i, j, k, l) I(h, i, j, k, l) (χ ) h!i! (i + j l)!(l + h j)! I(i + j l, l + h j, j, k, l) (χ 2) i!j! (j + k h)!(h + i k)! I(j + k h, h + i k, k, l, h) (χ 3) j!k! (k + l i)!(i + j l)! I(k + l i, i + j l, l, h, i) (χ 4) k!l! (l + h j)!(j + k h)! I(l + h j, j + k h, h, i, j) (χ 5) l!h! (h + i k)!(k + l i)! I(h + i k, k + l i, i, j, k) (χ 6) h!i!j! (k + l i)!(h + i k)!(i + j l)! I(k + l i, h + i k, k, l, i + j l) (χ 7 ) i!j!k! (j + k h)!(i + j l)!(l + h j)! I(j + k h, h, l, i + j l, l + h j) (χ 8 ) k!l!h! (k + l i)!(i + j l)!(l + h j)! I(i, k + l i, i + j l, l + h j, j) (χ 9 ) l!h!i! (h + i k)!(j + k h)!(l + h j)! I(h + i k, j + k h, l + h j, j, k) (χ 0 ) j!k!l! (h + i k)!(j + k h)!(k + l i)! I(h + i k, j + k h, h, i, k + l i) (χ ) h!i!j!k!l! (h + i k)!(k + l i)!(i + j l)!(l + h j)!(j + k h)! I(h + i k, k + l i, i + j l, l + h j, j + k h). (χ 2 ) Denote the integrals occurring in the above formulae by I (),..., I (2) respectively. Let M (), N () be the integers (2.9), associated with I () I(h, i, j, k, l), for every r,..., 2 let M (r), N (r) be the corresponding integers associated with I (r). Thus we have, for r 2, I (r) I(χ r (h), χ r (i), χ r (j), χ r (k), χ r (l))

19 Permutation group related to ζ(2) 4 where M (r) max{χ r (j + k h), χ r (k + l i), χ r (l + h j), χ r (h + i k), χ r (i + j l)}, N (r) max{τ r (M (r) ), min{τ 2 r (M (r) ), τ 3 r (M (r) )}, τ 4 r (M (r) )}, τ r χ r τ χ r. R e m a r k 3.. As we have already noted, the natural mapping Φ Φ considered above is an anti-isomorphism. Hence, for any χ Φ ϱ T, the integral (3.8) I(χϱ(h), χϱ(i), χϱ(j), χϱ(k), χϱ(l)) is obtained by applying to (3.9) I(χ(h), χ(i), χ(j), χ(k), χ(l)) the transformation ϱ T τ, σ corresponding to the permutation ϱ. Since the M, N defined by (2.9) are the same for each of ten integrals equivalent under the action of the transformation group T, the same integers M, N are associated with the integrals (3.8) (3.9). Therefore, for every r,..., 2 the pair M (r), N (r) is associated with each of the ten integrals obtained by applying to (h, i, j, k, l) the elements of the left coset χ r T. Thus if for any given χ Φ we replace I () I(h, i, j, k, l) with I () I(h, i, j, k, l ) I(χ(h), χ(i), χ(j), χ(k), χ(l)), accordingly each I (r) with I (r) I(χ r(h ), χ r(i ), χ r(j ), χ r(k ), χ r(l )), where we have χ r χχ r χ, I (r) I(χχ r (h), χχ r (i), χχ r (j), χχ r (k), χχ r (l)), the permutations χχ,..., χχ 2 are pairwise left-inequivalent mod T. Hence the 2 pairs M (r), N (r) associated with I (r) (r,..., 2) are the pairs M (r), N (r) in a different order. For the applications given in the next sections, we require the following Lemma 3.. There exists s, s 2, such that for every r,..., 2 we have M (r) M (s) N (r) N (s). P r o o f. Since M () max{j + k h, k + l i, l + h j, h + i k, i + j l}

20 42 G. Rhin C. Viola it is plain that M max M (r) r 2 M (2) max{k, i, l, j, h}, max{h, i, j, k, l, j + k h, k + l i, l + h j, h + i k, i + j l}. By Remark 3. we may assume with no loss of generality that M max M (r) i + j l, r 2 for otherwise we should replace I () I(h, i, j, k, l) with I(χ(h), χ(i), χ(j), χ(k), χ(l)) for a permutation χ Φ such that M χ(i + j l). From the expressions for M (),..., M (2) we see that each of the integers h, i, j, k, l, j + k h, k + l i, l + h j, h + i k, i + j l occurs exactly in six among M (),..., M (2). Let M (r ),..., M (r 6) be those containing the maximal integer i + j l in their expressions, whence let M M (r )... M (r 6) i + j l, N max{n (r ),..., N (r 6) } (we actually have {r,..., r 6 } {, 3, 5, 6, 0, }, but we do not need this information). In order to show that M, N is the maximal pair M (s), N (s) we are seeking, we must prove that N (r) N for any r r,..., r 6. If r r m, m 6, we know that N N (r m) max{τ rm (i + j l), τ 4 r m (i + j l)}, where τ rm (i + j l) τ 4 r m (i + j l) are (cyclically) consecutive to i + j l on either side in the expression for M (r m). If we pick the integers consecutive to i + j l in the expressions for M (r ),..., M (r 6), we see that (3.0) j + k h, h + i k, h, i, j, k N. Also, for any r r,..., r 6 the expression for M (r) contains exactly four among the integers (3.) j + k h, h + i k, h, i, j, k. If the integer Q r different from (3.) in the expression for M (r) satisfies Q r N, then N (r) M (r) N; if Q r > N then Q r M (r) by (3.0), N (r) max{τ r (Q r ), min{τ 2 r (Q r ), τ 3 r (Q r )}, τ 4 r (Q r )} is the maximum of three integers among (3.), whence N (r) (3.0). We conclude this section with two conjectures. N by

21 Permutation group related to ζ(2) 43 Conjecture 3.. Let h, i, j, k, l h, i, j, k, l be any non-negative integers. If I(h, i, j, k, l) I(h, i, j, k, l ), then there exists a permutation ϱ T τ, σ such that h ϱ(h), i ϱ(i), j ϱ(j), k ϱ(k), l ϱ(l). Conjecture 3.2. Let h, i, j, k, l h, i, j, k, l be any non-negative integers such that j + k h, k + l i, l + h j, h + i k, i + j l j + k h, k + l i, l + h j, h + i k, i + j l are also non-negative. If I(h, i, j, k, l) I(h, i, j, k, l ) Q, then there exists a permutation χ Φ ϕ, τ, σ such that h χ(h), i χ(i), j χ(j), k χ(k), l χ(l). 4. The p-adic valuation. Given five non-negative integers h, i, j, k, l such that j + k h, k + l i, l + h j, h + i k, i + j l are also nonnegative, we consider again the integrals I (),..., I (2) obtained by applying to (h, i, j, k, l) the permutations (3.7), for every r,..., 2 the pair M (r), N (r) associated with I (r), as is described in Section 3. By Remark 3. we may shall assume, with no loss of generality, that the maximal pair M (s), N (s) considered in Lemma 3. is obtained for s, for otherwise we should replace I () I(h, i, j, k, l) with b (r) n I () I(h, i, j, k, l ) I (s) I(χ s (h), χ s (i), χ s (j), χ s (k), χ s (l)). For r,..., 2 n, 2,... we define I (r) n I(χ r (h)n, χ r (i)n, χ r (j)n, χ r (k)n, χ r (l)n) a (r) n b (r) n ζ(2), where χ r is the rth permutation in the list (3.7). By Theorem 2.2 we have Z d M nd (r) N na (r) (r) n Z. For brevity, we shall omit the superscript r throughout. Thus, here in the sequel we abbreviate I n I(hn, in, jn, kn, ln) a n b n ζ(2), we denote again by M, N the integers (2.9). The transformation formula h!i! I(h, i, j, k, l) I(i + j l, l + h j, j, k, l) (i + j l)!(l + h j)!

22 44 G. Rhin C. Viola given in Section 3, corresponding to the permutation χ 2 ϕ, yields (hn)!(in)! I n a n b n ζ(2) ((i + j l)n)!((l + h j)n)! I(2) n (hn)!(in)! ((i + j l)n)!((l + h j)n)! (a(2) n b (2) n ζ(2)). It follows that (4.) ((i + j l)n)!((l + h j)n)!a n (hn)!(in)!a (2) n. Let d Mn d Nn a n A n d M (2) nd N (2) na (2) n A (2) n, whence A n, A (2) n Z. Multiplying (4.) by d Mn d Nn we get (4.2) ((i + j l)n)!((l + h j)n)!a n K(hn)!(in)!A (2) n, where d Mnd Nn K d M nd (2) N (2) n is an integer since M (2) M N (2) N. For any prime p any integer H > 0 we denote by v p (H) the p-adic valuation of H, i.e. the exponent of p in the factorization of H into prime powers. Define We have α p v p (((i + j l)n)!((l + h j)n)!) β p v p ((hn)!(in)!). v p (H!) m whence v p (H!) [H/p] if p > H. Since M max M (r) r 2 [ ] H p m, max{h, i, j, k, l, j + k h, k + l i, l + h j, h + i k, i + j l}, for any prime p satisfying (4.3) p > Mn we have Let [ ] [ ] (i + j l)n (l + h j)n α p + p p β p [ ] hn + p ω {n/p}, [ ] in. p

23 Permutation group related to ζ(2) 45 where {x} x [x] denotes the fractional part of x. For any integer H we have [ ] Hn n p H + Hω, p whence Therefore [ ] [ ] Hn n H + [Hω]. p p α p β p [(i + j l)ω] + [(l + h j)ω] [hω] [iω]. Lemma 4.. Let t, u, v, w R satisfy t + u v + w. Then P r o o f. We have whence [t] + [u] [v] [w]. [t] + [u] [v] [w] {v} + {w} {t} {u}, 2 < {t} {u} [t] + [u] [v] [w] {v} + {w} < 2. By Lemma 4. we have α p β p. Hence, removing from (4.2) the primes p > Mn dividing the factorials on both sides, we obtain (4.4) (p... p λ )P A n K(p... p λ )P A (2) n, where p,..., p λ are the distinct primes satisfying (4.3) for which α p β p, i.e. [hω] + [iω] < [(i + j l)ω] + [(l + h j)ω], p,..., p λ are the distinct primes satisfying (4.3) for which α p β p, i.e. (4.5) [(i + j l)ω] + [(l + h j)ω] < [hω] + [iω], P, P are products of primes Mn. From (4.4) we see that p,..., p λ divide A n. Therefore, any prime p satisfying (4.3) (4.5) divides A n. The above argument applies to each of the transformation formulae in Section 3 corresponding to the permutations χ 2, χ 3, χ 4, χ 5, χ 6. Thus we have proved the following

24 46 G. Rhin C. Viola Theorem 4.. Let ω {n/p}. Any prime p > Mn satisfying at least one of the following five inequalities: (4.6) divides A n d Mn d Nn a n. [(i + j l)ω] + [(l + h j)ω] < [hω] + [iω], [(j + k h)ω] + [(h + i k)ω] < [iω] + [jω], [(k + l i)ω] + [(i + j l)ω] < [jω] + [kω], [(l + h j)ω] + [(j + k h)ω] < [kω] + [lω], [(h + i k)ω] + [(k + l i)ω] < [lω] + [hω], We now consider the transformation formulae in Section 3 containing three factorials in the numerator three in the denominator, e.g. the formula corresponding to the permutation χ 7 ϕτ ϕ. With notation similar to the above, we have (4.7) ((k + l i)n)!((h + i k)n)!((i + j l)n)!a n L(hn)!(in)!(jn)!A (7) n, L d Mn d Nn /(d M (7) nd N (7) n) are inte- where A (7) n d M nd (7) N na (7) (7) n gers. For any prime p > Mn let whence, as before, γ p v p (((k + l i)n)!((h + i k)n)!((i + j l)n)!) δ p v p ((hn)!(in)!(jn)!), γ p δ p [(k + l i)ω] + [(h + i k)ω] + [(i + j l)ω] [hω] [iω] [jω] where V + V 2, V [(h + i k)ω] + [(k + l i)ω] [lω] [hω] V 2 [lω] + [(i + j l)ω] [iω] [jω] satisfy V m (m, 2) by Lemma 4.. Therefore 2 γ p δ p 2. Removing from (4.7) the primes p > Mn dividing the factorials, we now have (4.8) (p... p λ )(q... q µ ) 2 P A n L(p... p λ )(q... q µ )2 P A (7) n, where p,..., p λ ; q,..., q µ ; p,..., p λ ; q,..., q µ are the distinct primes

25 Permutation group related to ζ(2) 47 satisfying (4.3) for which γ p δ p, 2,, 2 respectively, P, P are products of primes Mn. From (4.8) we see that p,..., p λ q 2,..., q µ 2 divide A n. Therefore, any prime p > Mn for which γ p δ p < 0, i.e. (4.9) [(k + l i)ω] + [(h + i k)ω] + [(i + j l)ω] < [hω] + [iω] + [jω], divides A n, any prime p > Mn for which γ p δ p 2 is such that p 2 A n. Note that the condition γ p δ p 2 is equivalent to V V 2, i.e. to { [(h + i k)ω] + [(k + l i)ω] < [lω] + [hω], (4.0) [lω] + [(i + j l)ω] < [iω] + [jω]. It is easy to show that any prime p satisfying (4.9) must also satisfy at least one of (4.6), so that (4.9) yields no further primes besides those considered in Theorem 4.. Hence (4.9), as well as the similar inequalities given by the transformation formulae corresponding to the permutations χ 8, χ 9, χ 0, χ, can be disregarded. To prove this, let V k [(i + j l)ω] + [(l + h j)ω] [hω] [iω], V l [(j + k h)ω] + [(h + i k)ω] [iω] [jω], V h [(k + l i)ω] + [(i + j l)ω] [jω] [kω], V i [(l + h j)ω] + [(j + k h)ω] [kω] [lω], V j [(h + i k)ω] + [(k + l i)ω] [lω] [hω], whence γ p δ p [(k + l i)ω] + [(h + i k)ω] + [(i + j l)ω] [hω] [iω] [jω] 2 (V k + V l + V h V i + V j ). If (4.9) holds we have V k + V l + V h + V j < V i where, by Lemma 4., V k, V l, V h, V i, V j would be either 0 or if all the inequalities (4.6) were false. Thus V k V l V h V j 0, V i, whence γ p δ p 2 (V k +V l +V h V i +V j ) 2, a contradiction since γ p δ p is an integer. Hence (4.9) implies at least one of (4.6). From (4.0) the similar inequalities given by the transformation formulae corresponding to χ 8, χ 9, χ 0, χ we obtain the following Theorem 4.2. Let ω {n/p}. Any prime p > Mn satisfying at least one of the following five conditions:

26 48 G. Rhin C. Viola (4.) [(h + i k)ω] + [(k + l i)ω] < [lω] + [hω] [lω] + [(i + j l)ω] < [iω] + [jω], [(i + j l)ω] + [(l + h j)ω] < [hω] + [iω] [hω] + [(j + k h)ω] < [jω] + [kω], [(j + k h)ω] + [(h + i k)ω] < [iω] + [jω] [iω] + [(k + l i)ω] < [kω] + [lω], [(k + l i)ω] + [(i + j l)ω] < [jω] + [kω] [jω] + [(l + h j)ω] < [lω] + [hω], [(l + h j)ω] + [(j + k h)ω] < [kω] + [lω] [kω] + [(h + i k)ω] < [hω] + [iω], is such that p 2 divides A n d Mn d Nn a n. R e m a r k conditions: (4.2) 4.. If we apply the permutation σ to (4.) we get the [(l + h j)ω] + [(j + k h)ω] < [kω] + [lω] [lω] + [(i + j l)ω] < [iω] + [jω], [(k + l i)ω] + [(i + j l)ω] < [jω] + [kω] [kω] + [(h + i k)ω] < [hω] + [iω], [(j + k h)ω] + [(h + i k)ω] < [iω] + [jω] [jω] + [(l + h j)ω] < [lω] + [hω], [(i + j l)ω] + [(l + h j)ω] < [hω] + [iω] [iω] + [(k + l i)ω] < [kω] + [lω], [(h + i k)ω] + [(k + l i)ω] < [lω] + [hω] [hω] + [(j + k h)ω] < [jω] + [kω], which are easily seen to be equivalent to (4.), since the decomposition γ p δ p V + V 2, with V [(k + l i)ω] + [(i + j l)ω] [jω] [kω] V 2 [kω] + [(h + i k)ω] [hω] [iω], shows that the first of (4.) is equivalent to the second of (4.2), similarly for the other conditions. Hence Theorem 4.2, as well as Theorem 4., is invariant under the actions of τ σ. We finally consider the last transformation formula corresponding to the permutation χ 2. For any prime p > Mn let ε p v p (((h+i k)n)!((k +l i)n)!((i+j l)n)!((l +h j)n)!((j +k h)n)!)

27 Permutation group related to ζ(2) 49 whence ζ p v p ((hn)!(in)!(jn)!(kn)!(ln)!), ε p ζ p [(h + i k)ω] + [(k + l i)ω] + [(i + j l)ω] + [(l + h j)ω] + [(j + k h)ω] [hω] [iω] [jω] [kω] [lω] 2 (V k + V l + V h + V i + V j ). By Lemma 4. we have (V k + V l + V h + V i + V j ) 5 2. Since ε p ζ p is an integer, we get 2 ε p ζ p 2. If all the inequalities (4.6) are false we have ε p ζ p 2 (V k + V l + V h + V i + V j ) 0. Hence any prime p for which ε p ζ p < 0 satisfies at least one of (4.6). If ε p ζ p 2 we get V k + V l + V h + V i + V j 4 whence, by Lemma 4., four among V k, V l, V h, V i, V j are the remaining one is 0. If e.g. V k V l V h V j, V i 0, we obtain γ p δ p 2 (V k + V l + V h V i + V j ) 2, we have seen that this condition is equivalent to (4.0), i.e. to the first of (4.). In a similar manner, any other case where ε p ζ p 2 implies one of (4.). Hence the transformation formula corresponding to χ 2 yields no further primes besides those given by Theorems We now follow Hata s method (see [7], Section 5) to define an integer D n such that D n d Mn d Nn D n a n Z. To this purpose, however, we employ our Theorems in place of Hata s analysis of the p-adic valuation of binomial coefficients. Let Ω be the set of ω [0, ) satisfying at least one of the inequalities (4.6), let Ω be the set of ω [0, ) satisfying at least one of the conditions (4.). Since each of (4.) implies one of (4.6), we have Ω Ω. Clearly Ω Ω are unions of finitely many intervals [α q, β q ), where each α q or each β q is a rational number whose denominator divides at least one of the integers h, i, j, k, l, j + k h, k + l i, l + h j, h + i k, i + j l. Again we assume that the maximal pair M (s), N (s) considered in Lemma 3. is M () M, N () N. Lemma 4.2. If ω Ω then ω /M. If ω Ω then ω /N. P r o o f. Since M max{j + k h, k + l i, l + h j, h + i k, i + j l} max M (r) r 2 max{h, i, j, k, l, j + k h, k + l i, l + h j, h + i k, i + j l}, for any ω satisfying ω < /M we have ω < M min { h, i, j, k, l },

28 50 G. Rhin C. Viola whence [hω] [iω] [jω] [kω] [lω] 0. Therefore all the inequalities (4.6) are false, we get ω Ω. Take now any ω satisfying ω < /N. As in the proof of Lemma 3. we may assume with no loss of generality that M i+j l. For otherwise we should replace h, i, j, k, l with τ m (h), τ m (i), τ m (j), τ m (k), τ m (l) respectively, where m is such that M τ m (i + j l), this leaves unchanged the pair M, N as well as the sets Ω Ω. By (3.0) we have ω < { N min h, i, j, }, k whence Therefore [hω] [iω] [jω] [kω] 0. [lω] + [(i + j l)ω] [iω] + [jω] 0, [hω] + [(j + k h)ω] [jω] + [kω] 0, [(j + k h)ω] + [(h + i k)ω] [iω] + [jω] 0, [(k + l i)ω] + [(i + j l)ω] [jω] + [kω] 0, [kω] + [(h + i k)ω] [hω] + [iω] 0. Thus all the conditions (4.) are false, whence ω Ω. We now define n p> Mn {n/p} Ω p, n p> Mn {n/p} Ω p (n, 2,...), where p denotes a prime. From Theorems it follows that (4.3) n n A n. If p n we have ω {n/p} Ω, whence ω /M by Lemma 4.2. Therefore n/p ω /M, p Mn, p d Mn. Hence (4.4) n d Mn. Similarly, if p n then ω {n/p} Ω, whence n/p ω /N again by Lemma 4.2, we have p Nn, p d Nn. Hence (4.5) n d Nn. Thus, if we define (4.6) D n d Mn d Nn n, n

29 Permutation group related to ζ(2) 5 we have D n Z D n d Mn d Nn by (4.4) (4.5), D n a n n Z n by (4.3). In order to get from this an irrationality measure of ζ(2), we recall the following well-known lemma (see [8], Lemma 4): Lemma 4.3. Let α R, let (r n ), (s n ) be sequences of integers satisfying lim n n log r n s n α R lim sup n n log s n S for some positive numbers R S. Then Since A n µ(α) S R +. D n I n D n a n D n b n ζ(2) with D n, b n D n a n integers, we apply Lemma 4.3 with Thus we need lim n n log D n, r n D n a n, s n D n b n. lim n n log I n, lim sup n n log b n. Since d Mn d Nn exp((m + N)n + o(n)) by the prime number theorem, we have by (4.6) lim n n log D n M + N lim n n log n lim n n log n. As is shown by Hata (see [6] [7]), the last two limits exist can be easily computed by means of ψ(x) Γ (x) Γ (x), the logarithmic derivative of the Euler gamma-function. The result is \ lim n n log \ n dψ(x), lim n n log n dψ(x). Ω Ω Therefore ( \ \ ) (4.7) lim n n log D n M + N dψ(x) + dψ(x). Ω Ω

30 52 G. Rhin C. Viola 5. The irrationality measure of ζ(2). We now evaluate lim n n log I n, lim sup n n log b n, under the assumption that the integers h, i, j, k, l, j + k h, k + l i, l + h j, h + i k, i + j l are all strictly positive. Since \ \ ( x h ( x) i y k ( y) j ) n dx dy I n I(hn, in, jn, kn, ln) ( xy) i+j l xy, it is plain that (5.) lim n n log I n max log xh ( x) i y k ( y) j 0<x,y< ( xy) i+j l. Note that the maximum exists, since by the assumption h, i, j, k, l > 0 the function (5.2) f(x, y) xh ( x) i y k ( y) j ( xy) i+j l is continuous on the compact square [0, ] [0, ], zero on the boundary positive inside. From Lemma 2.6 we get, for any ϱ, ϱ 2 > 0 any n, n log b n log ϱh ( + ϱ ) i (/ϱ + ϱ 2 ) k ( + /ϱ + ϱ 2 ) j (ϱ ϱ 2 ) i+j l log ( + ϱ ) i ( + ϱ ϱ 2 ) k ( + ϱ + ϱ ϱ 2 ) j ϱ j+k h. (ϱ ϱ 2 ) i+j l Thus, denoting ϱ u, ϱ ϱ 2 v, we obtain lim sup n n log b n min log ( + u)i ( + v) k ( + u + v) j u,v>0 u j+k h v i+j l. Again by our assumption the minimum exists, since the function ( + u) i ( + v) k ( + u + v) j (5.3) u j+k h v i+j l is easily seen to be large outside a compact subset of {u > 0, v > 0}. If we make the change of variables u x, v xy, the function (5.3) becomes Therefore (5.4) lim sup n ( ) h+i+j+k+l xh ( x) i y k ( y) j ( xy) i+j l. n log b n min x,y<0 xy> log x h ( x) i y k ( y) j ( xy) i+j l.

31 Permutation group related to ζ(2) 53 It is easy to see that there are exactly two stationary points of the function (5.2) satisfying x( x)y( y) 0, since the system is equivalent to (5.5) y h x i (i + j l)y + x xy k y j (i + j l)x + 0 y xy (l + h j)(h + i k)x 2 + ((l + h j)(j + k h) h(h + i k) + j(i + j l))x h(j + k h) 0, (h + i k)x + k h (h + i k j)x + j + k h. Let (x 0, y 0 ) (x, y ) be the solutions of (5.5), with 0 < x 0 < x < 0. From (5.) (5.4) we obtain (5.6) (5.7) lim n n log I n log xh 0( x 0 ) i y0 k ( y 0 ) j ( x 0 y 0 ) i+j l, lim sup n n log b n log x h ( x ) i y k ( y ) j ( x y ) i+j l We can summarize some of the preceding results as follows. Let h, i, j, k, l > 0 be integers such that j + k h, k + l i, l + h j, h + i k, i + j l > 0, such that the pair M M (), N N () defined by (2.9) satisfies (5.8) M (r) M, N (r) N (r,..., 2), where M (r) max{χ r (j + k h), χ r (k + l i), χ r (l + h j),. χ r (h + i k), χ r (i + j l)}, N (r) max{τ r (M (r) ), min{τ 2 r (M (r) ), τ 3 r (M (r) )}, τ 4 r (M (r) )} is the pair associated with the integral I (r) I(χ r (h), χ r (i), χ r (j), χ r (k), χ r (l)), as is described in Section 3. Let Ω be the set of ω [0, ) satisfying at least one of the inequalities (4.6), let Ω be the set of ω [0, ) satisfying at least one of the conditions (4.). Let f(x, y) xh ( x) i y k ( y) j ( xy) i+j l, let (x 0, y 0 ) (x, y ) be the stationary points of f(x, y) satisfying x( x)y( y) 0, i.e. the solutions of (5.5), with 0 < x 0, y 0 < x, y < 0, let ψ(x) Γ (x)/γ (x). Denote

32 54 G. Rhin C. Viola c 0 log f(x 0, y 0 ), c log f(x, y ), ( \ \ ) c 2 M + N dψ(x) + dψ(x). Ω Ω Then, by (4.7), (5.6), (5.7) Lemma 4.3, we get the following Theorem 5.. With the above notation assumptions we have µ(ζ(2)) c + c 2 + c 0 + c, c 0 c 2 c 0 c 2 provided that c 0 > c 2. We now consider some special cases of Theorem 5., by making three different choices for h, i, j, k, l which yield good irrationality measures of ζ(2). We first consider Hata s integral \ \ ( x 5 ( x) 5 y 4 ( y) 4 ) n dx dy (5.9) ( xy) 2 xy, through which Hata proved first µ(ζ(2)) < (see [7]), later µ(ζ(2)) < ([7], Addendum) by applying to (5.9) our transformation στ. With our notation, the integral (5.9) is I(5n, 5n, 4n, 4n, 7n). The pair M M (), N N () associated with (5.0) I () I(5, 5, 4, 4, 7) is M 8, N 6, does not satisfy the technical assumption (5.8), since it is easy to check that the maximal pair M (s), N (s) considered in Lemma 3. is now M (s) 8, N (s) 7, is obtained for s 6 or s (this justifies the exceptional treatment of the interval [ 7, 6) in Hata s Addendum [7]). Therefore, in order to apply Theorem 5. we must rearrange the pairs M (r), N (r) (r,..., 2) by replacing (5.0) e.g. with I () I (6) I(6, 6, 5, 4, 4). Thus we choose h 6, i 6, j 5, k 4, l 4, whence j+k h 3, k+l i 2, l+h j 5, h+i k 8, i+j l 7, the pair M 8, N 7 defined by (2.9) satisfies (5.8). The set Ω is the union of the intervals [ 6, [ 2), 8, ) [ 6, 3 6, ) [ 4 7, 4, [ 8) 5, 2 7, ) [ 4 3, 5 6, ) [ 3, 5 4, [ 7) 7, 3 7, 7) 8, [ 2, ) [ 7 3, 9 6, [ 8), 5 8, ) [ 2 3, 6, ) [ 2 7, 5 7, ) [ 0 3, 4, [ 6) 5, 6 7, ) [ 8 9, 3 4, 8) 7, whence \ dψ(x) , Ω

33 Permutation group related to ζ(2) 55 the set Ω is the union of the intervals [ 4, ) [ 3, 7, ) [ 2 3, 3 4, ) [ 2 9, 3 8, [ 3) 5, 7 6, 9) 4, [ 4 7, ) [ 7 2, 9 4, ) [ 7, 3 6, ) [ 4 7, 7 8, ) [ 5 7, 5 6, 7) 6, whence \ dψ(x) , Ω so c Also c , c By Theorem 5. we obtain µ(ζ(2)) < As a second example, we make the somewhat simpler choice whence h 9, i 8, j 7, k 7, l 9, j +k h 5, k +l i 8, l+h j, h+i k 0, i+j l 6. The pair M, N 0 satisfies (5.8). The set Ω is the union of the intervals [ 9, ) [ 5, 2 9, ) [ 3, 2 7, ) [ 2 5, 3 7, ) [ 2, 5 9, ) [ 7, 2 3, ) [ 7 0, 5 7, ) [ 9, 6 7, ) 0, whence \ dψ(x) , Ω the set Ω is the union of the intervals [ 7, ) [ 6, 4 9, ) [ 5, 4 7, ) [ 3 5, 7 9, ) [ 4 5, 8 9, 0) 9, whence \ dψ(x) , Ω so c Here c , c Theorem 5. gives µ(ζ(2)) < Our final example is whence h 2, i 2, j 4, k 4, l 3, j+k h 6, k+l i 5, l+h j, h+i k 0, i+j l 3. The pair M 6, N 5 satisfies (5.8). The set Ω is the union of the intervals [ 4, [ 0), 7, [ 5), 3 4, ) [ 3, 2 7, [ 6) 5, 3, ) [ 2 5, 5 2, ) [ 7 5, 2, ) 6, [ 4 7, ) [ 5 8, 9 4, ) [ 7 0, 5 7, ) [ 5, 3 4, [ 6) 3, 5 6, ) [ 7 8, 2, 6) 5,

34 56 G. Rhin C. Viola whence \ dψ(x) , Ω the set Ω is the union of the intervals [ 2, [ ), 6, [ ) 2, 5 4, [ ) 4, 3 7, [ 6) 7, 7 2, ) [ 3 5, 4, ) [ 4 5, 6 7, ) [ 3 5, 3 4, 5) 4, whence \ dψ(x) , Ω so c Also c , c By Theorem 5. we now have as is stated in the introduction. µ(ζ(2)) < , References [] F. B e u k e r s, A note on the irrationality of ζ(2) ζ(3), Bull. London Math. Soc. (979), [2] A. C. Dixon, On a certain double integral, Proc. London Math. Soc. (2) 2 (905), 8 5. [3] R. Dvornicich C. Viola, Some remarks on Beukers integrals, in: Colloq. Math. Soc. János Bolyai 5, Budapest, 987, [4] A. Erdélyi et al., Higher Transcendental Functions (Bateman Manuscript Project), Vol., McGraw-Hill, New York, 953. [5] M. H a t a, Legendre type polynomials irrationality measures, J. Reine Angew. Math. 407 (990), [6], Rational approximations to π some other numbers, Acta Arith. 63 (993), [7], A note on Beukers integral, J. Austral. Math. Soc. A 58 (995), [8] G. Rhin C. Viola, On the irrationality measure of ζ(2), Ann. Inst. Fourier (Grenoble) 43 (993), [9] E. A. Rukhadze, A lower bound for the approximation of ln 2 by rational numbers, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 987 (6), (in Russian). Département de Mathématiques Dipartimento di Matematica U.F.R. MIM Università di Pisa Université de Metz Via Buonarroti 2 Ile du Saulcy 5627 Pisa, Italy Metz cedex, France viola@dm.unipi.it rhin@poncelet.univ-metz.fr Received on in revised form on (2864)