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pure and applied mathematics quarterly 2, 2 (2006) 435-463

Transcendence of periods: the state of the art

Michel Waldschmidt 1

(2006)

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.

1 :	Institut de Mathématiques de Jussieu (IMJ)
	CNRS : UMR7586 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

Domaine

:

Mathématiques/Théorie des nombres

Mots Clés : 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).

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Contributeur : Michel Waldschmidt <>

Soumis le : Jeudi 27 Août 2009, 10:58:21

Dernière modification le : Jeudi 27 Août 2009, 11:24:54