# [seqfan] Counting modules over the ring of integers in quadratic number fields

Neil Sloane njasloane at gmail.com
Sun Apr 3 06:24:22 CEST 2022

```This is another case where we had trouble because the original author (of
sequences A038540, A038541) did not provide a program.
I asked Don Zagier for help, and he kindly explained how to solve the
problem in general. You can see what he said - and his Pari program - in
A352550.
The two sequences I mentioned were for discriminants 40 and -20,
and Don pointed out that we should really have similar sequences for all
the smaller discriminants. I have added them, see A352550-A352567.

Don's solution makes use of the Dedekind zeta functions for these 20 number
fields, so I also made sure they too were in the OEIS.  There were in the
OEIS, yes, but none of them were called by that name (which I have now

Dedekind zeta functions for imaginary quadratic number fields of
discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324
<https://oeis.org/A002324>, A002654 <https://oeis.org/A002654>, A035182
<https://oeis.org/A035182>, A002325 <https://oeis.org/A002325>, A035179
<https://oeis.org/A035179>, A035175 <https://oeis.org/A035175>, A035171
<https://oeis.org/A035171>, A035170 <https://oeis.org/A035170>,
respectively.

Dedekind zeta functions for real quadratic number fields of discriminants
5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are *A035187
<https://oeis.org/A035187>*, A035185 <https://oeis.org/A035185>, A035194
<https://oeis.org/A035194>, A035195 <https://oeis.org/A035195>, A035199
<https://oeis.org/A035199>, A035203 <https://oeis.org/A035203>, A035188
<https://oeis.org/A035188>, A035210 <https://oeis.org/A035210>, A035211
<https://oeis.org/A035211>, A035215 <https://oeis.org/A035215>, A035219
<https://oeis.org/A035219>, A035192 <https://oeis.org/A035192>,
respectively.

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com

```