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Orthogonality and Irrationality

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Orthogonality and Irrationality ORTHOGONALITY AND IRRATIONALITY PETER BORWEIN, TAMAS´ ERDELYI,´ AND JOHN ZHANG This version was printed on November 3, 2013 Abstract. The fairly recent proofs of the irrationality of log 2,π2, ζ(3) and various values of polylogarithms are re-examined by using a particular family of orthogonal forms. This allows us to give a uniform and arguably more natural treatment of these results. 1. Introduction Ap´ery’s astonishing proof of the irrationality of ζ(3) amounts to showing that 0 < d3 a ζ(3) b 0 | k k − k| −→ where bk is an integer, k i + k 2 k 2 a := k k i i=0 X and d := lcm 1, 2,...,k . k { } Here and throughout the paper lcm denotes the least common multiple. This is entertainingly proved in van der Poorten [15] and given a very elegant treatment by Beukers [4], [5]. In [5] Beukers puts the proof in the context of Pad´e approximants. Many, maybe most, irrationality proofs may be based on approximation by Pad´e approximants and related orthogonal polynomials ([8], [9], [14]). Sometimes this is very natural, as with e, where the Pad´eapproximants generate convergents of the simple continued fraction. This is, however, an exceptional case. It is the intention of this paper to try to put the well known proofs of the irrationality of the constants log 2, ζ(2), and ζ(3), as well as certain values of poly- logarithms into the framework of orthogonality. This reproduces results of Ap´ery, Alladi and Robinson, and Chudnovsky in a unified fashion ([15], [2], [10]). In [7], the authors considered orthogonalization of the system xλ0 , xλ1 , xλ2 ,... , { } and from this point of view orthogonalization of the system x0, x0,...,x0 , x1, x1,...,x1 ,... { } n times n times | {z } | {z } 1991 Mathematics Subject Classification. 11J72, 41A20, 42C05. Key words and phrases. Riemann zeta function, polylogarithm, irrationality, orthogonality, Pad´eapproximant, M¨untz-Legendre polynomial. Research of the first author is supported, in part, by the NSERC of Canada. Research of the second author is supported, in part, by the NSF of the USA under Grant No. DMS-9024901, and conducted while an NSERC International Fellow at Dalhousie University. 1 2 PETERBORWEIN,TAMAS´ ERDELYI,´ AND JOHN ZHANG is very natural to study (the repeated indices are taken to be in a limiting sense). This leads to orthogonal functions that generalize Legendre polynomials and are of the form n−1 j Aj,l(x) log x j=0 X where each Aj,l(x) is a polynomial of degree l. Legendre polynomials are closely tied to irrationality questions concerning log- arithms ([2], [6, Chapter 11]) and higher order analogues prove to be the basis for zn dealing with the irrationality of the polylogarithm ([12]). nm n We argue that the irrationality proofs flow quiteX naturally from this point of view. A nice feature of this approach is that the non-vanishing of the estimates, which is essential for the proofs, comes very easily from the orthogonality. This paper is organized in the following way. First we produce a contour integral which generalizes the Legendre polynomials and consider the properties of this function. We then specialize this form for particular cases in order to prove the irrationality of log 2, ζ(2), ζ(3), and some values of the polylogarithm in general. The approach to ζ(2), ζ(3), and log(2) eventually reproduces the estimates of Ap´ery and Alladi and Robinson [2]. In the case of ζ(2) and ζ(3) we end up with the integrals of Beukers [4]. The results for irrationality of polylogarithms are very similar to those of Chudnovsky [10] but the estimates are different. In all cases, the estimate allow for inequalities of the form p 1 α > − q qc for all integers p and q > 0 for some c> 0 . So in all cases the stronger conclusion, that the numbers in question are not Liouville is possible. 2. The Contour Integral The orthogonalization of repeated monomials can be expressed by contour inte- grals. Theorem 1. Let k, l, m, and n be positive integers satisfying (l + 1)n km 1. Define − ≥ n k m (n 1)! (l!) j=1(t + j) t F (x; k,l; m,n) := − m l x dt 2πi (k!) γ (t j)n I Q j=0 − where γ is any simple contour containing the polesQ at t =0, 1, . , l. Then n−1 F (x; k,l; m,n) = ( 1)nl A (x) logj x − j,l j=0 X where Ai,l := Ai,l,k,m,n is a polynomial of degree l. So, l i + k m l n A (x) := ( 1)ni xi n−1,l − k i i=0 X ORTHOGONALITY AND IRRATIONALITY 3 and l k l i + k m l n m n A (x) := ( 1)ni (n 1) xi n−2,l − k i − i + j − i j i=0 j=1 X X j=0X,j6=i − Furthermore, this function enjoys the orthogonality relations 1 F (x; k,l; m,n)xi logj xdx =0, i =0, 1,...,k 1, j =0, 1,...,m 1. − − Z0 The polynomial An−1,l has integer coefficients. If k = l and n 2 then the polynomial d A also has integer coefficients (d := lcm 1, 2,...,k≥ ). k n−2,l k { } Proof. The representation of F is just the evaluation of the integral at the poles t = 0, 1,... , l by the Residue Theorem. The orthogonality conditions follow by interchanging the order of integrations and observing the behaviour as the contour is allowed to become arbitrarily large. The fact that An−1,l has integer coefficients is obvious from the above forms. To show that An−2,l has integer coefficients whenever k = l and n 2, we need to observe that if p is a prime and α is a positive integer so that k ≥pα k + i then i+k ≤ ≤ p divides k . This is straightforward from Euler’s formula for the largest power of a prime dividing a factorial. It is interesting to note that the orthogonality relation stated above encompasses an m-dimensional orthogonality conditions on the m-dimensional cube Im. That is, m x si x F (πm( ); k,l; m,n) xi d =0, si 0, 1,...,k 1 m ∈{ − } I i=1 Z Y m where πm(x) := xi and where x := (x1, x2,...,xm). Here and throughout i=1 Y 1 1 1 g(x)dx = g(x)dx1dx2 dxm. m ··· ··· ZI Z0 Z0 Z0 This can be deduced from the following theorem. Theorem 2. We have 1 1 n−1 F (πn(x))dx = F (u) log u du. n (n 1)! | | ZI − Z0 Proof. It is straightforward to verify by using the change of variables 1 u := xy, v := (x2 y2) 2 − that the double integral can be transformed as 1 1 1 F (xy)dxdy = F (u) log udu. − Z0 Z0 Z0 In the general case, the proof follows recursively. Various specializations of the parameters give us the results we are looking for. 4 PETERBORWEIN,TAMAS´ ERDELYI,´ AND JOHN ZHANG 3. Irrationality of log 2 and ζ(3) For these cases we set l = k and n = m in Theorem 1, so k m (m 1)! j=1(t + j) t F (x; k, k; m,m)= − k x dt 2πi γ (t j)m I Qj=0 − which are orthogonal in the sense that Q 1 F (x; k, k; m,m)F (x; k′, k′; m,m)dx =0 Z0 for any two distinct positive integers k and k′. Also dj F = 0, j =0, 1,...,m 2 dxj |x=1 − and dm−1 F = (m 1)!. dxm−1 |x=1 − The above equations can be easily verified by using the results of Theorem 1. When we set m = 1 and m = 2 these are in fact the polynomials needed in the proof of the irrationality of log 2 and ζ(3) respectively. Both proofs are very similar and we outline them below. Theorem 3. We have 1 F (x; k, k;1, 1) 0 < dx = A ( 1) log 2 + R (√2 1)2k log 2 1+ x | 0,k − k|≤ − Z0 where A ( 1) is an integer, R is a rational number, and d R is an integer 0,k k k k (d := lcm 1−, 2,...,k ). k { } Proof. We decompose the rational function as follows F (x; k, k;1, 1) A ( 1) A (x) A ( 1) = ( 1)k 0,k − + 0,k − 0,k − . 1+ x − 1+ x 1+ x The first term of the right-hand side integrates to A ( 1) log 2 while the second 0,k − integrates to a rational number which can be integralized by the factor dk introduced by the integration process. To establish the inequality we note that k i + k k dk xk(1 x)k A (x)= ( 1)i xi = − 0,k − k i dxk k! i=0 X which is the shifted Legendre polynomial of degree k. By making use of the above Rodrigues’ formula for these polynomials, and integrating by parts, we find 1 F (x; k, k;1, 1) 1 xk(1 x)k 1 dx 0 < dx = − dx (√2 1)2k 1+ x (1 + x)k+1 ≤ 1+ x · − Z0 Z0 Z0 since x (1 x) 0 < − (√2 1)2 1+ x ≤ − for every x (0, 1). ∈ ORTHOGONALITY AND IRRATIONALITY 5 The irrationality of log 2 now simply follows by multiplying both sides of the (1+δ)k inequality by dk, which by the prime number theorem is of O(e ) for any δ > 0.
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