Ternary Quadratic Forms and Eta Quotients
Ternary Quadratic Forms and Eta Quotients
复制标题
DOI：
10.4153/cmb-2015-044-3
复制
发表时间：
2015-12
期刊：
影响因子：
--
通讯作者：
K. Williams
中科院分区：
文献类型：
--
作者：
K. Williams
设$\Eta\Left(z\Right)\，\Left(z\，\in\，\mathbb{C}，\，\OPERATATOR NAME{Im}\Left(z\Right)\，>；\，0\Right)$表示DedeKind ETA函数。我们使用最近的乘积求和公式，结合整数不可表示的某些三元二次型的条件，显式地给出了十个ETA商$$f\Left(z\Right)，：=\，{{\Eta}^{a\Left({{m}_{1}}\Right)}}\，\Left({{m}_{1}}z\Right)\，.\，.{{\eta}^{a\Left({{m}_{r}}\Right)}}\，\Left({{m}_{r}}z\Right)\，=\，\sum\Limits_{n=1}^{\inty}{c\Left(n\right){{2\pi inz}}，\z\，\in，\mathbb{C}，\，操作员名称{Im}\Left(z\right)\，>；使得在无穷多个不重叠的算术级数中，对所有正整数$n$，傅立叶系数$c\Left(n\Right)$都为零。例如，如果$f\Left(z\Right)\，=\，{{\Eta}^{4}}\Left(z\Right){{\Eta}^{9}}\Left(4z\Right){{\Eta}^{-2}}\Left(8z\Right)$我们有$c\Left(n\right)\，=\，0$对于每个算术级数${16k\，+\，14\}}_{k\ge 0}}，{64k\，+\，56\}}_{k\ge 0}}，\，{{256k，\，224\}_{k\ge 0}}，\，{1024k\，+\，869\}}_{k\ge 0}}，\，.\，.$
Abstract Let $\eta \left( z \right)\,\left( z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0 \right)$ denote the Dedekind eta function. We use a recent product-to-sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotients $$f\left( z \right)\,:=\,{{\eta }^{a\left( {{m}_{1}} \right)}}\,\left( {{m}_{1}}z \right)\,.\,.\,.\,{{\eta }^{a\left( {{m}_{r}} \right)}}\,\left( {{m}_{r}}z \right)\,=\,\sum\limits_{n=1}^{\infty }{c\left( n \right){{e}^{2\pi inz}},\,\,\,z\,\in \,\mathbb{C},\,\operatorname{Im}\left( z \right)\,>\,0,}$$ such that the Fourier coefficients $c\left( n \right)$ vanish for all positive integers $n$ in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if $f\left( z \right)\,=\,{{\eta }^{4}}\left( z \right){{\eta }^{9}}\left( 4z \right){{\eta }^{-2}}\left( 8z \right)$ we have $c\left( n \right)\,=\,0$ for all $n$ in each of the arithmetic progressions ${{\{16k\,+\,14\}}_{k\ge 0}},\,{{\{64k\,+\,56\}}_{k\ge 0}},\,{{\{256k,\,224\}}_{k\ge 0}},\,{{\{1024k\,+\,869\}}_{k\ge 0}},\,.\,.\,.$