A Riccati differential equation and free subgroup numbers for lifts of modulo prime powers

It is shown that the number f λ of free subgroups of index 6λ in the modular group PSL 2 ( Z ) , when considered modulo a prime power p α 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 PSL 2 ( Z ) as a special case), and for a one-parameter family of lifts of the Hecke group H ( 4 ) = C 2 ⁎ C 4 . 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 PSL 2 ( Z ) and its lifts display rather complex behaviour modulo powers of 2 and 3.