Arithmetic properties derived from coefficients of certain eta quotients

[摘要] For a positive integer k, let $$ F (q)^{k}:= \prod_{n \geq 1} \frac{(1-q^{n})^{4k}}{(1+q^{2n})^{2k}} = \sum_{n\geq 0} \frak{a}_{k} (n)q^{n} $$ be the eta quotients. The coefficients $\frak{a}_{1} (n)$ can be interpreted as a certain kind of restricted divisor sums. In this paper, we give the signs and modulo values for $\frak{a}_{1} (n)$ and $\frak{a}_{2} (m)$ and calculate several convolution sums involving $\frak{a}_{k} (n)$.

[学科分类] 电力

[关键词] Restricted divisor functions;Eta quotient;Convolution sums;q -series