The statistics of the continued fraction digit sum

The statistics of the digits of a continued fraction, also known as partial quotients, have been studied at least since the time of Gauss. The usual measure m on the open interval (0,1) gives a probability space U. Let ak, k 1 be integer-valued random variables which take (0,1) to the kth partial quotient or digit in the continued fraction expansion = 1/(a1 + 1/(a2 + )). Let Sr = Sr() = rk=1 ak. It is well known that although there is an average value for log ak, each ak, let alone each Sr., has infinite expected value or first moment.

Overview