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Examples of multiplicative $$\eta$$-products. (English) Zbl 0597.10025
With the Dedekind function $$\eta$$ (z) and suitable finite sets of integer pairs (r,t) one can form products $$f(z)=\prod \eta (tz)^ r$$ which have an expansion $$f(z)=\sum a_ n e^{2\pi inz}$$ (n$$\geq 0)$$. A classification of $$\eta$$-products satisfying $$a_ 1\neq 0$$ and $$a_ 1a_{mn}=a_ ma_ n$$ for $$(m,n)=1$$ was given by D. Dummitt, H. Kisilevsky and J. McKay [Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)] in the case of positive exponents r. Allowing also negative exponents the present paper discusses further examples of such multiplicative products. These arise from certain polynomials $$\prod (z^ t-1)^ r$$ of degree 24 which are characteristic polynomials of automorphisms of the Leech lattice or are found by a computer search.
Reviewer: H.-G.Quebbemann

##### MSC:
 11F11 Holomorphic modular forms of integral weight 11H06 Lattices and convex bodies (number-theoretic aspects)