Zaremba’s conjecture and sums of the divisor function

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, cx.cr] for a/m has c¡ < B for i = 1, 2,., r, where B is a small absolute constant (say 5 = 5). Zaremba was only able to prove an estimate of the form c, < C log m for an absolute constant C. His first proof only applied to the case where m isa 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.