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Multiple series connected to Hoffman’s conjecture on multiple zeta values. (English) Zbl 1186.11051
A challenging problem in the arithmetic study of the so-called multiple zeta values (MZVs) \[ \zeta(s_1,\dots,s_l) =\sum_{n_1>\dots>n_l\geq1}\frac1{n_1^{s_1}\dotsb n_l^{s_l}}, \qquad s_1,\dots,s_l\in\mathbb Z_{>0}, \quad s_1\geq2, \] is determining all linear (and polynomial) relations for them over \(\mathbb Q\). One of the conjectures, due to M. Hoffman [J. Algebra 194, No. 2, 477–495 (1997; Zbl 0881.11067)], suggests that the MZVs with entries \(s_j\in\{2,3\}\) form a basis of the vector space spanned over \(\mathbb Q\) by all MZVs. This is shown to be equivalent to the linear independence of these restricted MZVs over \(\mathbb Q\). The only known way to approach the desired independence is constructing ‘small’ linear forms in \(1\) and \(\zeta(s_1,\dots,s_l)\) with \(s_j\in\{2,3\}\) with rational coefficients. The approach has been already successfully applied in the arithmetic study of the odd zeta values [K. Ball and T. Rivoal, Invent. Math. 146, No. 1, 193–207 (2001; Zbl 1058.11051)]; W. Zudilin, [Izv. Math. 66, No. 3, 489–542 (2002; Zbl 1114.11305)] and of the multiple zeta values (with non-strict inequalities in the index summation) whose entries are 2 only [V. N. Sorokin, Sb. Math. 187, No. 12, 1819–1852 (1996; Zbl 0876.11035)]. General results due to S. Zlobin [Math. Notes 77, No. 5-6, 630–652 (2005; Zbl 1120.11030)] suggest to construct such linear forms by means of multiple hypergeometric series \[ \sum_{k_1\geq\dots\geq k_l\geq1} \frac{P(k_1,\dots,k_l)}{(k_1+r_1)_{n_1+1}^3\dotsb(k_l+r_l)_{n_l+1}^3}, \qquad (k+r)_{n+1}=\prod_{j=0}^n(k+r+j), \tag{1} \] for polynomials \(P\) with rational coefficients and non-negative integers \(r_1,\dots,r_l\) and \(n_1,\dots,n_l\) subject to certain conditions. The main goal of the article is to control of what MZVs from Hoffman’s conjecture appear in the decomposition of these series under natural constraints on the polynomial \(P\) (that is, stability of the polynomial under the actions of certain finite groups). This is the matter already discussed in the recent joint work of the author [J. Cresson, S. Fischler, and T. Rivoal, J. Reine Angew. Math. 617, 109–151 (2008; Zbl 1227.11097)]. General results of the present article address more general series than in (1) (exponents 3 may be replaced by different positive integers); the complete control of the MZVs of maximal length (or depth) \(l\) and a partial control of the MZVs of length \(l-1\) appearing in the decomposition of (1) is provided. An application of Theorem 3.7 in the article and the above cited results of Zlobin is the following statement (Theorem 1.7): If \(n\), \(r_1\) and \(r_2\) are non-negative integers, \(r_1\geq r_2+n+1\), and a polynomial \(P\in{\mathbb Q}[k_1,k_2]\) has degree at most \(3n+1\) with respect to each variable and satisfies \(P(k_2+r_2-r_1,k_1+r_1-r_2)=-P(k_1,k_2)\), then the series \[ \sum_{k_1\geq k_2\geq1} \frac{P(k_1,k_2)}{(k_1+r_1)_{n+1}^3(k_2+r_2)_{n+1}^3} \] represents a linear combination of \(1\), \(\zeta(2)\), \(\zeta(3)\) and \(\zeta(2,3)-\zeta(3,2)\) with coefficients in \(\mathbb Q\).
It is definitely desirable to relate the series treated in the article under review to the known classes of multiple hypergeometric series. This is a program done in the single-variable case by the reviewer (see [J. Théor. Nombres Bordeaux 15, No. 2, 593–626 (2003; Zbl 1156.11326); J. Théor. Nombres Bordeaux 16, No. 1, 251–291 (2004; Zbl 1156.11327)]) in connection with the above mentioned result of Ball and Rivoal.

MSC:
11M32 Multiple Dirichlet series and zeta functions and multizeta values
33C70 Other hypergeometric functions and integrals in several variables
11J72 Irrationality; linear independence over a field
11J91 Transcendence theory of other special functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M41 Other Dirichlet series and zeta functions
33B30 Higher logarithm functions
33C20 Generalized hypergeometric series, \({}_pF_q\)
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