Zaremba’s conjecture and sums of the divisor function

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.