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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

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