Let n(z) (ϵ) denote the Dedekind eta function. We use a recent product- To-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotients f (z) := na(m1)(mr z) ... na(mrZ) = ∞ ∑ n=1 c(n)eπ2inz, z eϵ C, Im(z) > 0, n = 1 such that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if f(z) = n4 (z)n9(4z)n-2(8z) we have c(n) = 0 for all n in each of the arithmetic progressions {16k + 14}>0, {64k + 56}k>0, {256k + 224}k>0, {1024k + 896}k>0, ....

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Keywords Dedekind eta function, Eta quotient, Product-to-sum formula, Ternary quadratic form, Vanishing of fourier coef-ficients
Persistent URL dx.doi.org/10.4153/CMB-2015-044-3
Journal Canadian Mathematical Bulletin
Citation
Williams, K.S. (2015). Ternary quadratic forms and eta quotients. Canadian Mathematical Bulletin, 58(4), 858–868. doi:10.4153/CMB-2015-044-3