# What is the relationship between these two notions of “period”?

The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where $\mathcal{Z}$ is the $\mathbb{Q}$-span of the set of multiple zeta values (of positive integer arguments). My picture of mixed Tate motives is not very clear, and I would like to be able to relate their periods to something I understand better.

There is a survey article of Kontsevich and Zagier which defines a period as a complex number whose real and imaginary parts are given by convergent integrals of rational functions with rational coefficients, over domains in $\mathbb{R}^n$ cut out by finitely many polynomial inequalities with rational coefficients.

What is the relationship between the set of periods of mixed Tate motives over $\mathbb{Z}$ and the set of periods in the sense of Kontsevich/Zagier? Does one of these sets contain the other?

I would be interested to see examples of periods of one kind which are not periods of the other.

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Multiple zeta values can be defined by iterated integrals (see e.g. arxiv.org/pdf/1102.1310v2.pdf). It follows that periods of mixed Tate motives are periods in the sense of Kontsevich and Zagier (maybe you need to invert $2\pi i$). –  François Brunault Apr 7 '13 at 19:26
See also math.unice.fr/~brunov/GdT/… for a more detailed explanation. –  François Brunault Apr 7 '13 at 19:34
@François Brunault: But how do you invert $2\pi i$? Is this (now known to be) a period in the sense of Kontsevich--Zagier? In their paper they (if I remember well) mention specifically the (equivalent) question whether $1/\pi$ is a period or not. So, to me for the inclusion this is the point. –  quid Apr 7 '13 at 21:13
@quid: You're right that it is only conjectured that $1/(2\pi i)$ is not a period. Anyway, it is necessary to invert $2\pi i$ to have a valid statement because the period of the Tate motive $\mathbf{Q}(n)$ is $(2\pi i)^n$. –  François Brunault Apr 8 '13 at 6:45
This explains that it has become rather standard to consider the ring of extended periods $\widehat{\mathcal{P}} := \mathcal{P}[\frac{1}{2\pi i}]$. –  François Brunault Apr 8 '13 at 6:47