## On McKay’s conjecture.(English)Zbl 0548.10018

Let $$\eta(z)$$ be the Dedekind $$\eta$$-function. For any set of integers $$g=(k_ 1,...,k_ s)$$, $$k_ 1\geq k_ 2\geq...\geq k_ s\geq 1$$, we associate the $$\eta$$-product defined by $$\eta_ g(z)=\eta (k_ 1z)\eta (k_ 2z)...\eta (k_ sz).$$ J. McKay, D. S. Dummit and H. Kisilevsky [Multiplicative products of $$\eta$$-functions, preprint] considered $$\eta$$-products associated to all sets $$g$$ such that $$\sum^{s}_{i=1}k_ i=24$$. These functions have Fourier expansion of the form $$\eta_ g(z)=\sum^{\infty}_{n=1}a(n)q^ n, q=\exp (2\pi iz)$$. By computing $$a(n)$$ for small $$n$$, they found all sets $$g$$ whose $$\eta$$-products have multiplicative Fourier coefficients. They also found a certain combinatorial condition on $$k_ i$$ which just all the above sets satisfy.
In this paper, we use the theory of new forms to give an intrinsic proof of their results and also give a real meaning of their combinatorial condition.

### MSC:

 11F11 Holomorphic modular forms of integral weight
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### References:

 [1] DOI: 10.2969/jmsj/02620362 · Zbl 0273.14007 [2] DOI: 10.2969/jmsj/02610056 · Zbl 0266.12009 [3] M24 and certain automorphic forms, preprint [4] Multiplicative products of {$$\eta$$}-functions, preprint [5] DOI: 10.1007/BF01390245 · Zbl 0369.10016
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