Ternary Quadratic Forms and Eta Quotients

Canadian mathematical bulletin

Kenneth S. Williams¹ 

[1] Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6

关键词: Dedekind eta function;    eta quotient;    ternary quadratic forms;    vanishing of Fourier coefficients;    product-to-sum formula;   

DOI  :  10.4153/CMB-2015-044-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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 explicitly10 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^{2pii 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 ${16k+14}_{k geq 0}$, ${64k+56}_{k geq 0}$, ${256k+224}_{kgeq 0}$, ${1024k+896}_{k geq 0}$, $ldots$.

【 授权许可】

Unknown