Period (number theory)

A number \alpha is a period if it can be expressed as an integral of the form :\alpha = \int_{P(x_1,\ldots,x_n)\ge 0}Q(x_1,\ldots,x_n)\ \mathrm{d}x_1\ldots\mathrm{d}x_n where P is a polynomial and Q a rational function on \mathbb{R}^n with rational coefficients. ==Properties and motivation==

The periods are intended to bridge the gap between the well-behaved algebraic numbers, which form a class too narrow to include many common mathematical constants and the transcendental numbers, which are uncountable and apart from very few specific examples hard to describe. The latter are also not generally computable. The ring of periods \mathcal P lies in between the fields of algebraic numbers \mathbb \overline{Q} and complex numbers \mathbb C (ie \mathbb \overline{Q}\sub \mathcal P\sub \mathbb C) and is countable. The periods themselves are all computable, and in particular definable. Periods include some of those transcendental numbers, that can be described in an algorithmic way and only contain a finite amount of information. ==Numbers known to be periods==

The following numbers are among the ones known to be periods: ==Open questions==

Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods". Kontsevich and Zagier conjectured that, if a period is given by two different integrals, then each integral can be transformed into the other using only the linearity of integrals (in both the integrand and the domain), changes of variables, and the Newton–Leibniz formula : \int_a^b f'(x) \, dx = f(b) - f(a) (or, more generally, the Stokes formula). A useful property of algebraic numbers is that equality between two algebraic expressions can be determined algorithmically. The conjecture of Kontsevich and Zagier would imply that equality of periods is also decidable: inequality of computable reals is known recursively enumerable; and conversely if two integrals agree, then an algorithm could confirm so by trying all possible ways to transform one of them into the other one. Further open questions consist of proving which known mathematical constants do not belong to the ring of periods. An example of a real number that is not a period is given by Chaitin's constant Ω. Any other non-computable number also gives an example of a real number that is not a period. It is also possible to construct artificial examples of computable numbers which are not periods. However there are no computable numbers proven not to be periods, which have not been artificially constructed for that purpose. It is conjectured that 1/π, Euler's number e and the Euler–Mascheroni constant γ are not periods. Kontsevich and Zagier suspect these problems to be very hard and remain open a long time. ==Extensions==

The ring of periods can be widened to the ring of extended periods \hat \mathcal P by adjoining the element 1/π. They also form a ring and are countable. It is \overline{\mathbb Q}\sub \mathcal P \sube \mathcal{EP}\sub \mathbb C . The following numbers are among the ones known to be exponential periods: == See also ==