The statistics of the continued fraction digit sum
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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 α ∈ (0,1) to the kth partial quotient or digit in the continued fraction expansion α = 1/(a1 + 1/(a2 + ⋯)). Let Sr = Sr(α) = Σrk=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.