Canadian mathematical bulletin | |
Ternary Quadratic Forms and Eta Quotients | |
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
【 预 览 】
Files | Size | Format | View |
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RO201912050577176ZK.pdf | 13KB | download |