[seqfan] Dirichlet g.f. for A000688
israel at math.ubc.ca
israel at math.ubc.ca
Fri Nov 7 00:25:31 CET 2014
A000688 "Number of Abelian groups of order n; number of factorizations of n
into prime powers" currently has two different Dirichlet generating
functions listed:
Sum_{n >= 1} a(n)/n^s = Product_{k >= 0} zeta(s+k) = Product_{n=0..oo}
Product_{p prime} (1-p^(-s-n))^(-1). [Kurokawa and Wakayama, Proc. Japan
Acad. 78 Ser A (2002) p. 126.] - R. J. Mathar, Apr 03 2012.
and
sum_{n >= 1} a(n)/n^s = product_{k >= 1} zeta(ks). - Álvar Ibeas, Nov 05
2014
They can't both be correct. It seems pretty clear to me that the first one
is wrong (although it is indeed quoted accurately from the Kurokawa and
Wakayama paper). For example, the coefficient of p^(-s) for prime p would
be 1/(1-p) rather than 1. Kurokawa and Wakayama say it is "well-known" that
what they call zeta_H(s) can be considered as a generating function of the
order of finite abelian groups. They refer to Zagier, D.: Zetafunktionen
und quadratische k ̈orper. Springer-Verlag, Berlin-Heidelberg(1981). I
haven't looked at Zagier.
I think Álvar's D.g.f. is correct, but I don't have a proof.
So, what to do?
Cheers,
Robert
More information about the SeqFan
mailing list