Abstract

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 a_k, k ≥ 1 be integer-valued random variables which take α in (0,1) to the k^th partial quotient or digit in the continued fraction expansion α = 1 ∕ (a₁ + 1 ∕ (a₂ + ⋯)). Let S_r = S_r(α) = ∑ _k=1^ra_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 expected value or first moment.

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