Arithmetic hypergeometric series.(English. Russian original)Zbl 1225.33008

Russ. Math. Surv. 66, No. 2, 369-420 (2011); translation from Usp. Mat. Nauk 66, No. 2, 163-216 (2011).
In this survey paper the author demonstrates how the arithmetic hypergeometric series link certain seemingly unrelated research areas, and explains the underlying arithmetic and analytical techniques. More specifically, he addresses the following directions: (1) arithmetic properties of the values of Riemann’s zeta function $$\zeta(s)$$ and its generalizations at integers $$s>1$$; (2) the arithmetic significance of Calabi-Yau differential equations and generalized Ramanujan-type series for $$\pi$$; (3) hypergeometric and special-function evaluations of Mahler measures.

MSC:

 33C20 Generalized hypergeometric series, $${}_pF_q$$ 11J82 Measures of irrationality and of transcendence 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11Y60 Evaluation of number-theoretic constants 33C75 Elliptic integrals as hypergeometric functions