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 m easure m on the open interval (0, 1) gives a probability space U. Let a(k), k greater than or equal to 1 be integer-valued random variables which take alpha is an element of (0; 1) to the k th partial quotient or digit in the continued fraction expansion alpha = 1/(a(1)+ 1/(a(2) + ...)). Let S-r = S -r(alpha) = Sigma(k=1)(r) a(k). It is well known that although there is an average value for log a(k), each a(k), let alone each S-r, has infinite exp ected value or first moment.