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Ternary Quadratic Forms and Eta Quotients

  Published:2015-07-13
 Printed: Dec 2015
  • Kenneth S. Williams,
    Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6
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Abstract

Let $\eta(z)$ $(z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0)$ 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 10 eta quotients \[ f(z):=\eta^{a(m_1)}(m_1 z)\cdots \eta^{{a(m_r)}}(m_r z)=\sum_{n=1}^{\infty}c(n)e^{2\pi i nz},\quad z \in \mathbb{C},\;\operatorname{Im}(z)\gt 0, \] such that the Fourier coefficients $c(n)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, it is shown that for $f(z)=\eta^4(z)\eta^{9}(4z)\eta^{-2}(8z)$ we have $c(n)=0$ for all $n$ in each of the arithmetic progressions $\{16k+14\}_{k \geq 0}$, $\{64k+56\}_{k \geq 0}$, $\{256k+224\}_{k \geq 0}$, $\{1024k+896\}_{k \geq 0}$, $\ldots$.
Keywords: Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, product-to-sum formula Dedekind eta function, eta quotient, ternary quadratic forms, vanishing of Fourier coefficients, product-to-sum formula
MSC Classifications: 11F20, 11E20, 11E25 show english descriptions Dedekind eta function, Dedekind sums
General ternary and quaternary quadratic forms; forms of more than two variables
Sums of squares and representations by other particular quadratic forms
11F20 - Dedekind eta function, Dedekind sums
11E20 - General ternary and quaternary quadratic forms; forms of more than two variables
11E25 - Sums of squares and representations by other particular quadratic forms
 

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