The statistics of the continued fraction digit sum

Doug Hensley

Vol. 192 (2000), No. 1, 103–120

DOI: 10.2140/pjm.2000.192.103

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 ∈ (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.

Milestones

Received: 16 January 1996

Revised: 3 June 1996

Published: 1 January 2000