[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.


sum_{n >= 1} a(n)/n^s = product_{k >= 1} zeta(ks). - Álvar Ibeas, Nov 05 

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?


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