TITLE: Is there an explicit example of a complex number which is not a period?
QUESTION [33 upvotes]: Kontsevich and Zagier proposed a definition of a period (see for example http://en.wikipedia.org/wiki/Ring_of_periods ). The set of periods is countable, so not all of $\mathbb{C}$. I heard a rumour today that there is now a known explicit complex number which is not a period; does anyone know if this is true, or have more details?

REPLY [26 votes]: Such a number is constructed in this article of M.Yoshinaga where it is proved that periods can be effectively approximated by elementary rational Cauchy sequences.