Well-poised hypergeometric transformations of Euler-type multiple integrals.

*(English)*Zbl 1065.11054In this very interesting article, the author first makes a connection between a certain multiple integral and a very-well-poised series of hypergeometric type. He does this in Theorem 2, which is too complicated to write here. Here is a simple interesting case: for all integer \(n\geq 0\),

\[ \begin{split} S_n:=-\sum_{n=1}^{\infty} \frac{d}{d t} \bigg( (2t+n)\bigg(\frac{(t-1) \cdots(t-n)(t+n+1)\cdots(t+2n)} {(t(t+1)\cdots(t+n))^2}\bigg)^2 \bigg)\bigg|_{t=n} \\ =\frac{(3n+1)!}{n!^3}\int_{[0,1]^5} \frac{\prod_{j=1}^5 x_j^n(1-x_j)^n dx_1dx_2dx_3dx_4dx_5} {Q(x_1,x_2,x_3,x_4,x_5)^{n+1}}, \end{split}\tag{*} \] where

\[ Q(x_1,x_2,x_3,x_4,x_5) =x_1(1-(1-(1-(1-x_2)x_3)x_4)x_5) +(1-x_1x_2x_3x_4x_5). \]

In the Introduction, the author indicates possible applications of his Theorem 2 to the irrationality theory of the Riemann zeta function at integers. For example, the applications of (*) are the following. Let \(d_n=\mathrm{lcm}\{1, 2, \cdots, n\}\); by partial decomposition of the summand of \(S_n\), it can be shown that \(S_n\) is equal to \(u_n\zeta(4)-v_n\), where \(d_nu_n\in\mathbb{Z}\) and \(d_n^5v_n\in\mathbb{Z}\). The sequences \(u_n\) and \(v_n\) have also been obtained independently by V. N. Sorokin [An algorithm for fast calculation of \(\pi^4\), preprint, 59 pages, Moscow (2002)] and by H. Cohen and G. Rhin [see H. Cohen, Accélération de la convergence de certaines récurrences linéaires. Sémin. Théor. Nombres 1980–1981, Exp. No. 16 (1981; Zbl 0479.10023)]. Thus, the situation is apparently in complete analogy with the following result due to F. Beukers [Bull. Lond. Math. Soc. 11, 268–272 (1979; Zbl 0421.10023): there exist two sequences \(a_n\) and \(b_n\) such that \(a_n\in\mathbb{Z}\), \(d_n^3b_n\in\mathbb{Z}\) and \[ I_n := \int_{[0,1]^3} \frac{\prod_{j=1}^3 x_j^n(1-x_j)^n dx_1 dx_2 dx_3}{(1-(1-(1-x_1)x_2)x_3)^{n+1}}=a_n\zeta(3)-b_n. \] As is well-known, Apéry’s theorem that \(\zeta(3)\) is irrational follows since \(d_n^3I_n\) tends to 0 as \(n\) tends to infinity. Unfortunately, \(d_n^5S_n\) does not tend to 0 as \(n\) tends to infinity, and the irrationality of \(\zeta(4)\) cannot be proved this way (of course, \(\zeta(4)=\pi^4/90\) is known to be irrational but by completely different methods).

However there is more to say. In fact, the series \(S_n\) satisfies the hypothesis of the reviewer’s “denominators conjecture” [in T. Rivoal, J. Théor. Nombres Bordx. 15, No. 1, 351–365 (2003; Zbl 1041.11051)] which, in this case, says that, for all \(n\geq0\), \(u_n\in\mathbb{Z}\) and \(d_n^4b_n\in\mathbb{Z}\): this is still not enough to prove that \(\zeta(4)\) is irrational. In another paper [Well-poised hypergeometric service for diophantine problems of zeta values, J. Théor. Nombres Bordx. 15, No. 2, 593–626 (2003; Zbl 1156.11326), Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordx. 16, No. 1, 251–291 (2004; Zbl 1156.11327)], the author also proposed a more general “denominators conjecture” for certain very-well-poised series (to which his theorem 2 applies) which would not only prove that \(\zeta(4)\) is irrational, but also would provide the best known measure of its irrationality. This motivates Theorem 3 of the paper, in which the author studies the stability under a certain group of permutations of the identity in Theorem 2: such groups are crucial tools in the works of G. Rhin and C. Viola on the optimal irrationality measure for \(\zeta(2)\) [Acta Arith. 77, No. 1, 23–56 (1996; Zbl 0864.11037), resp. Acta Arith. 97, No. 3, 269–293 (2001; Zbl 1004.11042)].

The author also proves an interesting functional generalisation of one of his previous theorems (recalled as Theorem 1 in this paper and proved in the article mentioned above), which could prove fruitful in the arithmetical study of zeta. Finally, these results are based on various manipulations of complex Barnes type integrals and can be thought of as generalisations of some identities in the very classical Cambridge Tract 32 of W. N. Bailey on “Generalized Hypergeometric Series” [London: Cambridge Univ. Press (1935; Zbl 0011.02303)].

\[ \begin{split} S_n:=-\sum_{n=1}^{\infty} \frac{d}{d t} \bigg( (2t+n)\bigg(\frac{(t-1) \cdots(t-n)(t+n+1)\cdots(t+2n)} {(t(t+1)\cdots(t+n))^2}\bigg)^2 \bigg)\bigg|_{t=n} \\ =\frac{(3n+1)!}{n!^3}\int_{[0,1]^5} \frac{\prod_{j=1}^5 x_j^n(1-x_j)^n dx_1dx_2dx_3dx_4dx_5} {Q(x_1,x_2,x_3,x_4,x_5)^{n+1}}, \end{split}\tag{*} \] where

\[ Q(x_1,x_2,x_3,x_4,x_5) =x_1(1-(1-(1-(1-x_2)x_3)x_4)x_5) +(1-x_1x_2x_3x_4x_5). \]

In the Introduction, the author indicates possible applications of his Theorem 2 to the irrationality theory of the Riemann zeta function at integers. For example, the applications of (*) are the following. Let \(d_n=\mathrm{lcm}\{1, 2, \cdots, n\}\); by partial decomposition of the summand of \(S_n\), it can be shown that \(S_n\) is equal to \(u_n\zeta(4)-v_n\), where \(d_nu_n\in\mathbb{Z}\) and \(d_n^5v_n\in\mathbb{Z}\). The sequences \(u_n\) and \(v_n\) have also been obtained independently by V. N. Sorokin [An algorithm for fast calculation of \(\pi^4\), preprint, 59 pages, Moscow (2002)] and by H. Cohen and G. Rhin [see H. Cohen, Accélération de la convergence de certaines récurrences linéaires. Sémin. Théor. Nombres 1980–1981, Exp. No. 16 (1981; Zbl 0479.10023)]. Thus, the situation is apparently in complete analogy with the following result due to F. Beukers [Bull. Lond. Math. Soc. 11, 268–272 (1979; Zbl 0421.10023): there exist two sequences \(a_n\) and \(b_n\) such that \(a_n\in\mathbb{Z}\), \(d_n^3b_n\in\mathbb{Z}\) and \[ I_n := \int_{[0,1]^3} \frac{\prod_{j=1}^3 x_j^n(1-x_j)^n dx_1 dx_2 dx_3}{(1-(1-(1-x_1)x_2)x_3)^{n+1}}=a_n\zeta(3)-b_n. \] As is well-known, Apéry’s theorem that \(\zeta(3)\) is irrational follows since \(d_n^3I_n\) tends to 0 as \(n\) tends to infinity. Unfortunately, \(d_n^5S_n\) does not tend to 0 as \(n\) tends to infinity, and the irrationality of \(\zeta(4)\) cannot be proved this way (of course, \(\zeta(4)=\pi^4/90\) is known to be irrational but by completely different methods).

However there is more to say. In fact, the series \(S_n\) satisfies the hypothesis of the reviewer’s “denominators conjecture” [in T. Rivoal, J. Théor. Nombres Bordx. 15, No. 1, 351–365 (2003; Zbl 1041.11051)] which, in this case, says that, for all \(n\geq0\), \(u_n\in\mathbb{Z}\) and \(d_n^4b_n\in\mathbb{Z}\): this is still not enough to prove that \(\zeta(4)\) is irrational. In another paper [Well-poised hypergeometric service for diophantine problems of zeta values, J. Théor. Nombres Bordx. 15, No. 2, 593–626 (2003; Zbl 1156.11326), Arithmetic of linear forms involving odd zeta values, J. Théor. Nombres Bordx. 16, No. 1, 251–291 (2004; Zbl 1156.11327)], the author also proposed a more general “denominators conjecture” for certain very-well-poised series (to which his theorem 2 applies) which would not only prove that \(\zeta(4)\) is irrational, but also would provide the best known measure of its irrationality. This motivates Theorem 3 of the paper, in which the author studies the stability under a certain group of permutations of the identity in Theorem 2: such groups are crucial tools in the works of G. Rhin and C. Viola on the optimal irrationality measure for \(\zeta(2)\) [Acta Arith. 77, No. 1, 23–56 (1996; Zbl 0864.11037), resp. Acta Arith. 97, No. 3, 269–293 (2001; Zbl 1004.11042)].

The author also proves an interesting functional generalisation of one of his previous theorems (recalled as Theorem 1 in this paper and proved in the article mentioned above), which could prove fruitful in the arithmetical study of zeta. Finally, these results are based on various manipulations of complex Barnes type integrals and can be thought of as generalisations of some identities in the very classical Cambridge Tract 32 of W. N. Bailey on “Generalized Hypergeometric Series” [London: Cambridge Univ. Press (1935; Zbl 0011.02303)].

Reviewer: Tanguy Rivoal (Grenoble)