Transcendence of periods: the state of the art

Michel Waldschmidt (1)

1 IMJ - Institut de Mathématiques de Jussieu

The set of real numbers and the set of complex numbers have the power of continuum. Among these numbers, those which are ``interesting'', which appear ``naturally'', which deserve our attention, form a countable set. Starting from this point of view we are interested in the periods as defined by M.~Kontsevich and D.~Zagier. We give the state of the art on the question of the arithmetic nature of these numbers: to decide whether a period is a rational number, an irrational algebraic number or else a transcendental number is the object of a few theorems and of many conjectures. We also consider the approximation of such numbers by rational or algebraic numbers.

Mots clés: multiple zeta values (MZV); Periods; transcendental numbers; irrationality; integrals; series; Diophantine approximation; irrationality measures; transcendence measures; measures of algebraic independence; Gamma function; Beta function; zeta function; multiple zeta values (MZV).

Domaines: Théorie des nombres [math.NT]

Michel Waldschmidt. Transcendence of periods: the state of the art. Pure and Applied Mathematics Quarterly, 2006, 2 (2), pp.435-463. ⟨hal-00411301⟩