In this paper we study the set of values at algebraic points of analytic continuations of -functions (in the sense of Siegel). This subring of contains values of elliptic integrals, multiple zeta values, and values at algebraic points of generalized hypergeometric functions with rational coefficients. Its group of units contains non-zero algebraic numbers, , and (with such that , and such that exists and is non-zero). We prove that for any , both and can be written as , where is a -function with rational coefficients of which the radius of convergence can be made arbitrarily large. As an application, we prove that quotients of elements of are exactly the numbers which can be written as limits of sequences , where and are -functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for , and gives answers to questions asked by T. Rivoal in “Approximations rationnelles des valeurs de la fonction Gamma aux rationnels: le cas des puissances”, Acta Arith. 142 (2010), no. 4, 347–365.
Cite this article
Stéphane Fischler, Tanguy Rivoal, On the values of -functions. Comment. Math. Helv. 89 (2014), no. 2, pp. 313–341DOI 10.4171/CMH/321