Mobius aMUsements

[Avi Peretz]

Recently I contributed a comment about sequence A000203, which is sigma(n)
the sum of the divisors of n.
According to the reference of Lubotzky in sequence A000203:
A. Lubotzky, Counting subgroups of finite index, Proceedings of the
              St. Andrews/Galway 93 group theory meeting, Th. 2.1. LMS Lecture
Notes
              Series no. 212 Cambridge University Press 1995.

The number of sublattices of index n in Z^2 is sigma(n).

For a squarefree integer n:  phi+(N) =  sigma(n),  so which of these two is the
number of sublattices for general n ?

Regards,
Avi Peretz

John Conway wrote:

> On Fri, 4 May 2001, Marc LeBrun wrote:
>
>     [...]
>
> >    A1615:    1,3,4,6,6,12,8,12,12,18,12,24,14,24,24,24,18,36,20,36,32,36,...
> >
> > This sequence is described in the EIS as "Sublattices of index n in generic
> > 2-dimensional lattice; also index of GAMMA_0(n) in SL(2,z)."  (Anyone care
> > to roughly decipher this definition?)
>
>     Well, first let me say that since it's given by the formula
>
>         N  times the product of  1 + 1/p  (over primes  p|N )
>
> it's often called  phi+(N).
>
>     The 2-dimensional lattice  <V,W>  consists of all mectors of the form
> mV + nW,  (m,n  integers).  It has precisely  phi+(2) = 3  sublattices of
> index 2,  namely  <2V,W>, <V,2W> and  <V+W,2V> (which = <V+W,2W>),  and
> so for other indices.
>
>     SL2(Z) = Gamma  is the group of all 2x2 matrices
>
>          / a  b \
>         (        )    where  a,b,c,d  are integers  with  ad-bc = 1,
>          \ c  d /
>
> and Gamma0(N) is usually defined as the subgroup of this for which  N|c.
>
>    But conceptually  Gamma is best thought of as the group of (positive)
> automorphisms of a lattice  <V,W>,  its typical element taking
> V -> aV + bW,  W -> cV + dW,  and then  Gamma0(N)  can be defined as
> the subgroup consisting of the automorphisms that fix the sublattice
> <NV,W>  ( or should that be <V,NW> ? ) of index N.
>
>      Regards,  John Conway