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Mathematics of Computation

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Zaremba's conjecture and sums of the divisor function

Author: T. W. Cusick
Journal: Math. Comp. 61 (1993), 171-176
MSC: Primary 11J13; Secondary 11J25, 11J70
MathSciNet review: 1189517
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Abstract: Zaremba conjectured that given any integer $ m > 1$, there exists an integer $ a < m$ with a relatively prime to m such that the simple continued fraction $ [0,{c_1}, \ldots ,{c_r}]$ for a/m has $ {c_i} \leq B$ for $ i = 1,2 \ldots ,r$, where B is a small absolute constant (say $ B = 5$). Zaremba was only able to prove an estimate of the form $ {c_i} \leq C\log m$ for an absolute constant C. His first proof only applied to the case where m is a prime; later he gave a very much more complicated proof for the case of composite m. Building upon some earlier work which implies Zaremba's estimate in the case of prime m, the present paper gives a much simpler proof of the corresponding estimate for composite m.

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Article copyright: © Copyright 1993 American Mathematical Society