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Abstract: Let $\varphi\tau=\eta\tau+1-2^2-\sqrt{2\pi}e^\frac{\pii}{4}\eta\tau+1$ where $\eta\tau$ is the Dedekind eta-function. We showthat if $\tau 0$ is an imaginary quadratic number with $\mathrm{Im}\tau 0>0$and $m$ is an odd integer, then $\sqrt{m}\varphim\tau 0-\varphi\tau 0$ isan algebraic integer dividing $\sqrt{m}$. This is a generalization of Theorem4.4 given in B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan-s remarkableproduct of theta-functions, Proc. Edinburgh Math. Soc. 2 40 1997, no. 3,583-612. On the other hand, let $K$ be an imaginary quadratic field and$\theta K$ be an element of $K$ with $\mathrm{Im}\theta K>0$ which generatorsthe ring of integers of $K$ over $\mathbb{Z}$. We develop a sufficientcondition of $m$ for $\sqrt{m}\varphim\theta K-\varphi\theta K$ to become aunit.



Autor: Ja Kyung Koo, Dong Hwa Shin, Dong Sung Yoon

Fuente: https://arxiv.org/







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