I have just learned herehere that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the existence of a period whose inverse is not a period seems to be still open. Is this correct?
More generally: is it believed that the unit group of the ring of periods is bigger than the nonzero algebraic numbers?
