A Riccati differential equation and free subgroup numbers for lifts of PSL2(Z) modulo prime powers

It is shown that the number fλ of free subgroups of index 6λ in the modular group PSL2(Z), when considered modulo a prime power pα with p⩾5, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing PSL2(Z) as a special case), and for a one-parameter family of lifts of the Hecke group H(4)=C2⁎C4. All this is achieved by explicitly determining Padé approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main results complement previous work by Kauers and the authors (2012) [12,15], where it is shown, among other things, that the free subgroup numbers of PSL2(Z) and its lifts display rather complex behaviour modulo powers of 2 and 3.