The Statistics of the Continued Fraction Digit Sum
نویسندگان
چکیده
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 k partial quotient or digit in the continued fraction expansion α = 1/(a1 + 1/(a2 + · · · )). Let Sr = Sr(α) = ∑r k=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.
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