# Re David Wilson's question (response from M. Somos)

N. J. A. Sloane njas at research.att.com
Sun Jun 12 14:47:11 CEST 2005

Michael Somos, not a member of this list, but reachable at
somos at grail.cba.csuohio.edu, replied to me as follows (concerning
David Wilson's questions about multiplicative properties
of eta-functions):

(quote)

Neil,

> I'm sporadically working on Mitchell Harris's project to properly annotate
> as many as possible of the multiplicative sequences in the OEIS.

That is nice. So am I. There is no urgency. About those elliptic
functions, there is a known theory. I quote from Fine's book page 77
[Basic Hypergeometric Series and Applications]

32. Products with multiplicative series.  By a multiplicative series
we mean one of the form

(32.1)   A + B \sum_{N=1}^\infty C(N)q^N ,

where C(MN) = C(M)C(N)  for (M,N) = 1.  Such series which represent
modular functions of  \tau ( q = exp 2\pi i\tau)  have been studied
by Hecke [26], who has shown a deep-lying connection between the
arithmetic properties of the coefficients and the group- and function-
theoretic behaviro of the corresponding modular functions.

Andrews comments later on page 91 :

\S 32. The work in this section (as is pointed out) is closely
related to the extensive study made by Hecke of modular functions
with multiplicative coefficients. ...

Once I automate the process of submitting info about such sequences
I will do so. I don't have any proofs but some sequences of this type
are known to be multiplicative as Fine demonstrates. As one example
of what I will later do, there is a paper by Yves Martin about the
partitions of 24 and the corresponding eta-products. Many of those
are multiplicative, including A030199 which was mentioned. I will get
to it when I have time to automate the process of submitting info.

MR1376550 (97d:11070)
Martin, Yves(1-CA)
Multiplicative $\eta$-quotients. (English. English summary)
Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825--4856.
[...]
One consequence of the author's work is to describe precisely all
choices of $t_j$ and $r_j$ for which the corresponding
$\eta$-quotient (74 in all) is a holomorphic newform of integral
weight. Analogous results for $\eta$-products (i.e., where all
[...]
page 4852    Table I: Multiplicative \eta-quotients
[...]

I plan to submit info about all 74 of these series, but things take
time. I have to automate the process so I can deal with hundreds of
such sequences. Some already in OEIS, some not. Shalom, Michael

(end of quote)

NJAS