On a conjecture of Zaremba (Q580395)

For a natural number C and \(x\geq 1\) let \(N_ C(x)\) be the number of positive integers \(m\leq x\) such that there is an integer a with \(1\leq a<m\), \((a,m)=1\) and all partial quotients in the continued fraction expansion of a/m are at most C. The conjecture of Zaremba says \(N_ 5(x)=[x]\). We prove that \(N_ C(x) > (1/\sqrt{2C})x^{1/2(1-1/C^ 2)}.\)