Irrationality of certain numbers that contain values of the di- and trilogarithm. (English) Zbl 1105.11021

Hessami Pilehrood, Khodabakhsh; Hessami Pilehrood, Tatiana; Zudilin, Wadim

Math. Z. 254, No. 2, 299-313 (2006).

The authors prove that, for \(z \in \{1/2, 2/3, 3/4, 4/5\}\), at least one of the two numbers
\[ \text{Li}_2(z) + \log(1 - z) \log(z), \quad \text{Li}_3(z) + \frac 12 \log(1 - z) \log(z)^2, \]
is irrational, where \(\text{Li}_s(z) = \sum_{k=1}^{\infty} z^k/k^s\). In the case \(z = 1/2\), this improves the previous result of S. Fischler and the reviewer in [“Approximants de Padé et séries hypergéométriques équilibrées,” J. Math. Pures Appl. 82,No. 10, 1369–1394 (2003; Zbl 1064.11053)] that at least one of the three numbers \[ \text{Li}_2(1/2) + \log(1/2)^2, \quad \text{Li}_3(1/2) + \frac 12 \log(1/2)^3, \quad \text{Li}_4(1/2) + \frac 16 \log(1/2)^4 \] is irrational. The proof essentially uses two tools. Firstly, the method introduced by Fischler and the reviewer (in the above mentioned article): one can explicitly compute certain simultaneous Padé approximants for the families \((\text{Li}_s(z), s = 1,\dots, A)\) and \((\log^s(z), s = 1, \dots, A)\) at the point \(z = 0\) and \(z = 1\) respectively and combine them by multiplying one of the linear forms by \(\text{Li}_1(z) = -\log(1-z)\). Secondly, and this is the main new ingredient, the quality of the linear form is improved thanks to a now classical arithmetical method, which consists in finding and removing large common prime factors in the coefficients of the linear forms.

Reviewer: Tanguy Rivoal (Grenoble)

MSC:

11J72	Irrationality; linear independence over a field
11G55	Polylogarithms and relations with \(K\)-theory
33C20	Generalized hypergeometric series, \({}_pF_q\)
33C60	Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)

Citations:

Zbl 1064.11053

References:

[1]	Ball, K., Rivoal, T.: Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs. Invent. Math. 146, 193–207 (2001) · Zbl 1058.11051
[2]	Fischler, S., Rivoal, T.: Approximants de Padé et séries hypergéométriques équilibrées. J. Math. Pures Appl. 82, 1369–1394 (2003) · Zbl 1064.11053
[3]	Gutnik, L.A.: Irrationality of some quantities that contain {\(\zeta\)}(3) (Russian). Acta Arith. 42, 255–264 (1983) · Zbl 0474.10026
[4]	Hessami Pilehrood, T.G.: On the linear independence of vectors with polylogarithmic coordinates. Moscow Univ. Math. Bull. 54(6), 40–42 (1999) · Zbl 0983.11044
[5]	Lewin, L.: Polylogarithms and associated functions. Elsevier North Holland, New York 1981 · Zbl 0465.33001
[6]	Rhin, G., Viola, C.: On a permutation group related to {\(\zeta\)}(2). Acta Arith. 77, 23–56 (1996) · Zbl 0864.11037
[7]	Rhin, G., Viola, C.: The group structure for {\(\zeta\)}(3). Acta Arith. 97, 269–293 (2001) · Zbl 1004.11042
[8]	Rhin, G., Viola, C.: The permutation group method for the dilogarithm. Ann. Scuola Norm. Sup. Pisa 4(5), 389–437 (2005) · Zbl 1170.11316
[9]	Zudilin, W.: Arithmetic of linear forms involving odd zeta values. J. Théorie Nombres Bordeaux 16, 251–291 (2004) · Zbl 1156.11327